Introduction to Logarithms: Videos & Practice Problems

To solve equations involving exponents, we can use logarithms, which are the inverse of exponential functions. For example, if $2^x=216$, we can express this as $x=log\_2\left(216\right)$. Key properties include that $log\_b\left(b^x\right)=x$ and $log\_b\left(1\right)=0$. Understanding these concepts is crucial for evaluating logarithmic expressions without a calculator.

Logarithms Introduction

When solving polynomial equations, such as \(x^3 = 216\), the goal is to isolate \(x\) by performing the reverse operation. In this case, taking the cube root of both sides yields \(x = \sqrt[3]{216}\). However, when the variable \(x\) is in the exponent, as in \(2^x = 8\), we need to determine how many times \(2\) must be multiplied by itself to equal \(8\). This leads us to the conclusion that \(x = 3\).

For more complex equations like \(2^x = 216\), instead of multiplying \(2\) repeatedly, we can utilize logarithms. The logarithm is the inverse operation of exponentiation. To isolate \(x\), we take the logarithm of both sides, specifically using the same base as the exponential. Thus, we write:

\( \log_2(2^x) = \log_2(216) \)

Since the logarithm and the exponential share the same base, they cancel out, resulting in:

\(x = \log_2(216)\)

This expression is in logarithmic form, which indicates the power to which the base \(2\) must be raised to yield \(216\). This logarithmic form is equivalent to the original exponential equation \(2^x = 216\).

To convert between exponential and logarithmic forms, always start with the base of the exponential. For example, converting \(3^x = 81\) to logarithmic form involves identifying the base \(3\) and writing:

\(x = \log_3(81)\)

Conversely, to convert from logarithmic to exponential form, such as \(x = \log_4(64)\), we start with the base \(4\) and express it as:

\(4^x = 64\)

Let’s practice this conversion process with a few examples. Starting with \(x = \log_5(800)\), we convert it to exponential form:

\(5^x = 800\)

For \( \log_2(16) = 4\), we convert it to exponential form as:

\(2^4 = 16\)

Lastly, for \(10^x = 4500\), the conversion to logarithmic form gives:

\(x = \log_{10}(4500)\)

Notably, logarithms with base \(10\) are referred to as common logarithms and can simply be written as \(\log(4500)\). This common log has a dedicated button on calculators, making it convenient for evaluation.

Understanding these conversions and the properties of logarithms is essential for solving equations involving exponents efficiently.

The Natural Log

In mathematics, logarithms are essential tools for solving exponential equations. Among the most commonly used logarithms are the common logarithm, which is base 10, and the natural logarithm, which is base e. The natural logarithm is denoted as ln, a notation derived from the words "natural log." This special notation is important because it simplifies the representation of logarithmic expressions involving the base e.

When dealing with exponential equations, such as e^x = m, we can convert them into logarithmic form. The conversion follows the rule that if b^x = m, then log_b(m) = x. For the natural logarithm, this means that e^x = m can be rewritten as ln(m) = x. This transformation allows us to solve for the variable x in a straightforward manner.

For example, if we have x = ln(17), we can express this in exponential form as e^x = 17. Similarly, if we start with e^x = 4, we can rewrite it in logarithmic form as ln(4) = x. It is crucial to remember that whenever we encounter log base e, we should use the ln notation instead.

Understanding the natural logarithm and its properties is vital for solving various mathematical problems, especially in calculus and higher-level mathematics. The natural logarithm has its own dedicated button on calculators, making it easier to compute values quickly. As you practice converting between exponential and logarithmic forms, you'll become more comfortable with these concepts and their applications.

Change the following logarithmic expression to its equivalent exponential form.
$\log_4x=5$

• A. $4^{x}=5$
• B. $x^4=5$
• C. $4^5=x$
• D. $5^4=x$

Answer: (C)

Change the following logarithmic expression to its equivalent exponential form.
$x=\log9$

• A. $x=9$
• B. $9^{x}=10$
• C. $1^{x}=9$
• D. $10^{x}=9$

Answer: (D)

Change the following exponential expression to its equivalent logarithmic form.
$3^{x}=7$

• A. $\log_37=x$
• B. $\log_73=x$
• C. $\log_3x=7$
• D. $\log7=3^{x}$

Answer: (A)

Change the following exponential expression to its equivalent logarithmic form.
$e^9=x+3$

• A. $\log\left(x+3\right)=9$
• B. $\ln\left(x+3\right)=9$
• C. $\ln9=x+3$
• D. $\log_9x=e^3$

Answer: (B)

Evaluate Logarithms

Understanding logarithms is essential, as they serve as the inverse of exponential functions. This relationship allows us to evaluate logarithmic expressions without a calculator. For instance, consider the expression log₂(√[3]{2}). By recognizing that the cube root of 2 can be expressed as an exponent, we rewrite it as log₂(2^1/3). Utilizing the inverse property of logarithms, where log_b(b^x) = x, we find that this simplifies to 1/3.

Another important property is that the logarithm of a number to its own base equals 1. For example, log₂(2) simplifies to 1 because 2¹ = 2. Similarly, the logarithm of 1 in any base is always 0, as b⁰ = 1 for any base b.

To further illustrate these properties, consider the natural logarithm of 1, which is log_e(1) = 0. For the common logarithm, log(10) (which is log₁₀(10)), also equals 1. When evaluating log₅(1/5), we can rewrite 1/5 as 5^-1, leading to log₅(5^-1) = -1.

By mastering these properties, you can confidently evaluate logarithmic expressions without the need for a calculator, enhancing your mathematical skills and understanding of exponential relationships.

Evaluate the given logarithm.
$\log_77^{0.3}$

• A. 1.79
• B. 7
• C. 1
• D. 0.3

Answer: (D)

Evaluate the given logarithm.
$\frac32\log1$

• A. $\frac32$
• B. 0
• C. 1
• D. 10

Answer: (B)

Evaluate the given logarithm.
$\log_9\frac{1}{81}$

• A. 81
• B. 2
• C. – 2
• D. 9

Answer: (C)

Here’s what students ask on this topic:

What is the inverse relationship between exponentials and logarithms?

The inverse relationship between exponentials and logarithms is fundamental in mathematics. An exponential function, such as $a^x$, can be reversed using a logarithm. Specifically, if $a^x=y$, then the logarithmic form is $\log a_y=x$. This means that the logarithm gives the power to which the base must be raised to produce a given number. Understanding this relationship allows us to solve equations where the variable is an exponent, by converting them into a more manageable logarithmic form. This conversion is crucial in many fields, including science and engineering, where exponential growth and decay are common.

How do you convert an equation from exponential form to logarithmic form?

To convert an equation from exponential form to logarithmic form, follow these steps: Start with an exponential equation, such as $b^x=y$. Identify the base $b$, the exponent $x$, and the result $y$. The logarithmic form of this equation is $\log b_y=x$. This conversion is useful because it allows you to solve for the exponent $x$ when the base and the result are known. This process is essential in solving exponential equations, especially when the exponent is the unknown variable.

What are common logarithms and natural logarithms?

Common logarithms and natural logarithms are two frequently used types of logarithms. Common logarithms have a base of 10 and are denoted as $\log$. They are widely used in scientific calculations and are often found on calculators. Natural logarithms have a base of $e$ (approximately 2.718) and are denoted as $\ln$. They are particularly important in calculus and mathematical modeling of natural phenomena. Both types of logarithms simplify complex calculations by transforming multiplicative relationships into additive ones, making them invaluable tools in various fields of study.

How can you evaluate logarithms without a calculator?

Evaluating logarithms without a calculator involves understanding key properties of logarithms. For example, $\log b_b=\; 1$ because any number raised to the power of 1 is itself. Similarly, $\log b_1=\; 0$ because any number raised to the power of 0 is 1. Additionally, if the base of the logarithm and the argument are the same, such as $\log b_{b^x}$, the result is simply $x$. By applying these properties, you can simplify and evaluate logarithmic expressions without needing a calculator.

What is the process of converting between logarithmic and exponential forms?

Converting between logarithmic and exponential forms involves recognizing the relationship between the two. For a logarithmic equation $\log b_y=x$, the equivalent exponential form is $b^x=y$. To convert from logarithmic to exponential form, identify the base $b$, the exponent $x$, and the result $y$. Conversely, to convert from exponential to logarithmic form, use the same elements to express the equation as a logarithm. This conversion is essential for solving equations where the variable is in the exponent, allowing for easier manipulation and solution.