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. Converting between logarithmic and exponential forms, such as $y_{\log }=b_x$ and $x_b=b^y$, is essential. Properties like ${\log }_b(b^y)=y$ simplify evaluation. Rewriting roots as fractional exponents aids in solving logarithms, deepening understanding of exponents, bases, and arguments in polynomial and exponential contexts.

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\( means \)y = 3\( since \)2^3 = 8\(. However, when the number is not a perfect power of the base, such as \)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\(, which reads as "y equals log base b of x." Here, \)b\( is the base, and \)x\( is the argument of the logarithm. The fundamental definition states that \)y = \log_b x\( is equivalent to \)x = b^y\(. This equivalence highlights that a logarithm is essentially the exponent \)y\( to which the base \)b\( must be raised to yield \)x\(. For instance, \)y = \log_2 10\( means that \)y\( is the exponent to which 2 must be raised to get 10.

Logarithmic functions are the inverse of exponential functions. If an exponential function is written as \)f(x) = b^x\(, its inverse function is \)f^{-1}(x) = \log_b x\(. This inverse relationship is crucial because it allows solving for exponents in equations where the exponent is unknown. For example, to find the inverse of \)f(x) = 5^x\(, start by rewriting it as \)y = 5^x\(, then switch \)x\( and \)y\( to get \)x = 5^y\(. Solving for \)y\( involves rewriting the equation in logarithmic form: \)y = \log_5 x\(. Thus, the inverse function is \)f^{-1}(x) = \log_5 x$.

This interplay between exponential and logarithmic functions is foundational in algebra and calculus, enabling the solving of equations involving exponents and understanding growth and decay processes. Mastery of converting between exponential and logarithmic forms enhances problem-solving skills and deepens comprehension of function inverses.

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\(, the base is 3 and the exponent is 4. Converting this to logarithmic form involves making the base of the exponential the base of the logarithm, the result the argument of the logarithm, and the exponent the value of the logarithm: \)\log_3 81 = 4\(. This means that 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 conversion can be verified by evaluating the exponential expression.

In cases where the logarithm’s value is unknown, such as \)x = \log_5 800\(, the exponential form is written as \)5^x = 800\(. Conversely, an exponential expression like \)10^x = 4500\( can be rewritten in logarithmic form as \)\log_{10} 4500 = x$. This highlights the interchangeable nature of logarithmic and exponential expressions, emphasizing the importance of identifying the base and exponent to switch between forms accurately.

Mastering these conversions enhances problem-solving skills in algebra and deepens understanding of logarithmic functions, which are essential in various mathematical and scientific applications.

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

• A. ${\log }_{9}3=x$
• B. ${\log }_{3}x=9$
• C. ${\log }_{3}3=9$
• D. ${\log }_{{}_3}9=x$

Answer: (D)

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

• A. ${\log }_{6}216=-3$
• 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. ${\log }_{2}3.249=y$
• B. ${\log }_{2}y=3.249$
• 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. ${25}^{\frac{1}{2}}=5$
• 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 conceptual link between logarithms and exponents. Mastery of these concepts is foundational for solving more complex logarithmic and exponential problems in algebra and beyond.

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}}\(. This transformation allows the logarithmic expression to be rewritten as \)\log_4 \left(4^{\frac{1}{3}}\right)\(. Since the base of the logarithm and the base of the exponent are the same, the logarithm and the exponential function are inverse operations and effectively cancel each other out. This leaves the exponent as the result, so \)\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. $\frac{1}{2}$
• C. $y^2$
• D. $y^{\frac{1}{2}}$

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 inverses 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 convert between exponential and logarithmic forms, which is a fundamental skill in algebra and helps solve equations involving exponents.

How do you convert between exponential form and logarithmic form?

To convert from exponential form to logarithmic form, identify the base and the exponent in the exponential expression. For example, if you have $b_y=x$, the equivalent logarithmic form is $y=\log b_{}\left(x\right)$. The base of the exponential becomes the base of the logarithm, the exponent becomes the value of the logarithm, and the result of the exponential becomes the argument of the logarithm. Conversely, to convert from logarithmic to exponential form, rewrite $y=\log b_{}\left(x\right)$ as $b_y=x$. Mastering this conversion is essential for solving logarithmic and exponential equations.

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

There are several important properties of logarithms that simplify evaluation: 1) $\log b_{}\left(b^y\right)=y$ (Inverse Property) means a logarithm and an exponential with the same base cancel out. 2) $\log b_{}\left(b\right)=1$ because any base raised to the power 1 is itself. 3) $\log b_{}\left(1\right)=0$ since any base raised to the zero power equals 1. These properties allow you to evaluate logarithms without a calculator by rewriting arguments as powers of the base or using these fundamental rules.

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, apply the inverse property of logarithms: $\log 4_{}\left(4^{\frac{1}{3}}\right)=\frac{1}{3}$. This works because the logarithm and the exponential with the same base cancel out, leaving the fractional exponent. This technique is useful for simplifying and evaluating logarithms 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 variable is in the exponent. Without logarithms, isolating the exponent would be difficult or impossible. This inverse relationship also helps in converting between forms, simplifying expressions, and understanding the behavior of functions. It is foundational for topics like solving exponential growth and decay problems, analyzing polynomial degrees, and working with complex algebraic expressions involving exponents.