Introduction to Logarithmic Functions: Videos & Practice Problems

Logarithmic functions, inverses of exponential expressions, reveal the exponent a base must be raised to equal a number, expressed as $y_{\log }={\log }_b$ equivalent to $x=b^y$. Mastery of converting between logarithmic and exponential forms, focusing on the base and exponent, is essential. Properties like ${\log }_b=y$ enable evaluation without calculators, including rewriting roots as fractional exponents. These concepts underpin understanding of polynomials, exponents, and scientific notation.

Intro to Logarithms

Logarithmic functions are closely connected to exponential functions, serving as their inverse. Understanding logarithms begins with recognizing that a logarithm answers the question: "To what exponent must a base be raised to produce a given number?" For example, in the exponential equation \$2^y = 4\(, the exponent \)y\( is 2 because \)2^2 = 4\(. Similarly, \)2^y = 8\( implies \)y = 3\( since \)2^3 = 8\(. However, when the equation is \)2^y = 10\(, the exponent \)y\( is not an integer but lies between 3 and 4. This is where logarithms become essential.

The logarithmic notation expresses this relationship as \)y = \log_b x\(, which reads as "y equals the logarithm base b of x." Here, \)b\( is the base, matching the base in the exponential form, and \)x\( is the argument, the number we want to find the exponent for. The fundamental equivalence is:

\[x = b^y \quad \Longleftrightarrow \quad y = \log_b x\]

This means that the logarithm \)\log_b x\( gives the exponent \)y\( to which the base \)b\( must be raised to yield \)x\(. For instance, \)y = \log_2 10\( represents the exponent to which 2 must be raised to get 10.

Logarithmic and exponential functions are inverses of each other. Expressed in function notation, the logarithmic function \)f(x) = \log_b x\( is the inverse of the exponential function \)f(x) = b^x\(. This inverse relationship allows us to solve equations involving exponents by converting between exponential and logarithmic forms.

For example, to find the inverse of the exponential function \)f(x) = 5^x\(, start by rewriting it as \)y = 5^x\(. Swapping \)x\( and \)y\( gives \)x = 5^y\(. Solving for \)y\( involves applying the logarithm base 5:

\[y = \log_5 x\]

Thus, the inverse function is \)f^{-1}(x) = \log_5 x$. This process highlights how logarithms enable us to isolate exponents and solve exponential equations effectively.

Mastering the conversion between exponential and logarithmic forms is crucial for working with these functions, as it deepens understanding of their inverse nature and expands problem-solving capabilities in algebra and beyond.

Converting Between Exponential & Logarithmic Form

Understanding the relationship between logarithmic and exponential forms is fundamental in algebra. A logarithm represents the exponent to which a base must be raised to produce a given number. This relationship can be expressed algebraically as \(y = \log_b x\) being equivalent to \(x = b^y\), where b is the base, y is the exponent, and x is the result.

When converting from exponential to logarithmic form, focus on two key components: the base and the exponent. For example, in the exponential expression \$3^4 = 81\(, the base is 3 and the exponent is 4. Converting this to logarithmic form involves making the base of the exponential the base of the logarithm, the result the argument of the logarithm, and the exponent the value of the logarithm. Thus, it becomes \)\log_3 81 = 4\(, which means "3 raised to the power of 4 equals 81."

Similarly, when converting from logarithmic to exponential form, the base of the logarithm becomes the base of the exponent, the logarithm’s value becomes the exponent, and the argument becomes the result. For instance, given \)\log_2 16 = 4\(, the equivalent exponential form is \)2^4 = 16\(. This confirms that 2 raised to the 4th power equals 16.

Another example is \)\displaystyle x = \log_5 800\(, which converts to the exponential form \)5^x = 800\(. Here, the base 5 is raised to the unknown exponent x to yield 800. Conversely, an exponential expression like \)10^x = 4500\( can be rewritten in logarithmic form as \)\log_{10} 4500 = x$, indicating that the logarithm base 10 of 4500 equals the exponent x.

Mastering these conversions enhances problem-solving skills in algebra and deepens understanding of logarithmic and exponential functions. Remember, the base and exponent are the critical elements to identify when switching between forms, as the logarithm essentially answers the question: "To what power must the base be raised to produce this number?"

Rewrite the exponential equation as a logarithmic equation.
$3^x=9$

• A. $\log_93=x$
• B. $\log_3x=9$
• C. $\log_33=9$
• D. $\log_{_3}9=x$

Answer: (D)

Rewrite the exponential equation as a logarithmic equation.
$6^{-3}=\frac{1}{216}$

• A. $\log_6216=-3$
• B. $\log_{-3}\frac{1}{216}=6$
• C. $\log_6\frac{1}{216}=-3$
• D. $\log_{216}6=-3$

Answer: (C)

Rewrite the exponential equation as a logarithmic equation.
$2^y=3.249$

• A. $\log_23.249=y$
• B. $\log_2y=3.249$
• C. $\log_{3.249}y=2$
• D. $\log_{10}3.249=y$

Answer: (A)

Rewrite the logarithmic equation as an exponential equation.
${\log }_5\left(25\right)=2$

• A. $25^{\frac12}=5$
• B. $5^2=25$
• C. $2^5=25$
• D. $25^2=625$

Answer: (B)

Rewrite the logarithmic equation as an exponential equation.
${\log }_3\left(\frac{1}{27}\right)=-3$

• A. $-3^3=\frac{1}{27}$
• B. $-3^3=-27$
• C. $3^{-3}=\frac{1}{27}$
• D. $3^{-3}=-\frac{1}{27}$

Answer: (C)

Rewrite the logarithmic equation as an exponential equation.
${\log }_{4}x=1.5$

• A. $4^{x}=1.5$
• B. $4^{1.5}=x$
• C. $x^4=1.5$
• D. $x^{1.5}=4$

Answer: (B)

Basic Properties of Logarithms

Logarithms are fundamentally the inverse of exponential functions, which means they can be used to determine the exponent to which a base must be raised to produce a given number. This inverse relationship is key to evaluating logarithmic expressions without a calculator by applying specific properties derived from this connection.

One essential property is the inverse property of logarithms and exponentials: for a base b, the expression \(\log_b(b^x)\) simplifies directly to x. This occurs because the logarithm asks, "To what power must b be raised to get \(b^x\)?" The answer is clearly x, so the logarithm and the exponential cancel each other out. Similarly, raising the base to the logarithm of a number, \(b^{\log_b(x)}\), also returns x.

Another important property is that the logarithm of the base itself is always one, expressed as \(\log_b(b) = 1\). This is because any number raised to the power of one equals itself. Likewise, the logarithm of one, regardless of the base, is zero: \(\log_b(1) = 0\). This follows from the fact that any nonzero number raised to the zero power equals one.

These properties can be applied creatively to evaluate logarithms with arguments that are powers or fractions involving the base. For example, to evaluate \(\log_4(16)\), recognize that 16 can be rewritten as \$4^2\(. Using the inverse property, this simplifies to 2. Similarly, \)\log_5\left(\frac{1}{5}\right)\( can be rewritten as \)\log_5(5^{-1})$, which simplifies to -1 by the same property.

Understanding these logarithmic properties allows for efficient evaluation of logarithmic expressions without calculators, reinforcing the concept that logarithms and exponentials are inverse operations. This foundational knowledge is crucial for solving more complex logarithmic problems and deepening comprehension of exponential and logarithmic relationships.

Basic Properties of Logarithms Example 1

To evaluate logarithms that initially seem complex, it is helpful to rewrite expressions using exponent rules. For example, consider the logarithm with base 4 of the cube root of 4. The cube root can be expressed as an exponent: the cube root of 4 is equivalent to 4 raised to the power of one-third, or \$4^{\frac{1}{3}}\(. This allows the logarithmic expression to be rewritten as \)\log_4 \left(4^{\frac{1}{3}}\right)\(. Since the base of the logarithm and the base of the exponent are the same, the logarithm and the exponential function are inverse operations and effectively cancel each other out. This simplification leaves the exponent as the result, so \)\log_4 \left(4^{\frac{1}{3}}\right) = \frac{1}{3}$. This approach demonstrates the importance of recognizing how to convert roots into fractional exponents and applying the inverse property of logarithms to simplify and evaluate logarithmic expressions efficiently.

Evaluate the given expression.
$5^{{\log }_{5}12}$

• A. $12$
• B. $5.40$
• C. $41.94$
• D. $96.57$

Answer: (A)

Evaluate the given logarithm.
${\log }_y\sqrt{y}$

• A. $2$
• B. $\frac12$
• C. $y^2$
• D. $y^{\frac12}$

Answer: (B)

Evaluate the given logarithm.
${\log }_{x}1$

• A. $x$
• B. $1$
• C. $0$
• D. $2$

Answer: (C)

Here’s what students ask on this topic:

What is the relationship between logarithmic functions and exponential functions?

Logarithmic functions are the inverse of exponential functions. This means that if you have an exponential function expressed as $b_y=x$, the logarithmic function reverses this relationship and is written as $y=\log b_{}\left(x\right)$. Here, the logarithm tells you the exponent $y$ to which the base $b$ must be raised to get $x$. Understanding this inverse relationship is crucial because it allows you to solve for exponents in equations where the exponent is unknown, which is often difficult to do using only exponential form. This concept is foundational in algebra and helps in converting between exponential and logarithmic forms.

How do you convert between logarithmic form and exponential form?

To convert between logarithmic and exponential forms, focus on the base and the exponent. For an exponential equation like $b_{}{y}^{}=x$, the equivalent logarithmic form is $y=\log b_{}\left(x\right)$. The base $b$ remains the same, the exponent $y$ becomes the logarithm's value, and the result $x$ becomes the argument of the logarithm. Conversely, if you have a logarithmic expression like $y=\log b_{}\left(x\right)$, you can rewrite it as $b_{}{y}^{}=x$. Mastering this conversion is essential for solving logarithmic and exponential equations effectively.

What are the key properties of logarithms that help in evaluating logarithmic expressions?

There are several important properties of logarithms that simplify evaluation: 1) Inverse Property: $\log b_{}\left(b_{}{x}^{}\right)$ equals $x$ because the log and exponential with the same base cancel out. 2) Log of the Base: $\log b_{}\left(b\right)$ equals 1, since any base raised to the power 1 is itself. 3) Log of One: $\log b_{}\left(1\right)$ equals 0, because any base raised to the zero power is 1. These properties allow you to evaluate many logarithms without a calculator, especially when you can rewrite the argument as a power of the base.

How can you evaluate logarithms involving roots or fractional exponents?

To evaluate logarithms with roots, rewrite the root as a fractional exponent. For example, the cube root of 4 can be written as $4^{\frac{1}{3}}$. Then, if you have $\log 4_{}\left(4^{\frac{1}{3}}\right)$, the logarithm and exponential with the same base cancel, leaving just the fractional exponent $\frac{1}{3}$. This method leverages the inverse property of logarithms and exponentials, making it easier to evaluate logarithms involving roots or fractional powers without a calculator.

Why is it important to understand the inverse relationship between logarithms and exponentials in algebra?

Understanding that logarithms are the inverse of exponentials is crucial because it allows you to solve equations where the unknown is an exponent. Before learning logarithms, solving for the exponent in equations like $2_{}{y}^{}=10$ was difficult. Logarithms provide a way to rewrite such equations as $y=\log 2_{}\left(10\right)$, directly expressing the exponent. This inverse relationship also helps in finding inverse functions, converting between forms, and applying properties that simplify complex expressions. Mastery of this concept is foundational for higher-level algebra and calculus topics.