Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each exponential equation in logarithmic form. $$ 64=8^{2} $$

Step-by-Step Solution

Answer

The logarithmic form is \( \log_8 64 = 2 \).

Step 1: Identify the components

To convert the exponential equation into logarithmic form, first identify the components of the exponential equation. In this case, the exponential equation is given by: \[ 64 = 8^2 \]Here, 64 is the result (or the power), 8 is the base of the exponential expression, and 2 is the exponent.

Step 2: Convert to logarithmic form

The general form of an exponential equation is \( b^x = y \), and its logarithmic form is \( \log_b y = x \). Using the equation from Step 1 (\( 64 = 8^2 \)), rewrite it in logarithmic form: \[ \log_8 64 = 2 \]Here, 64 is converted to the logarithm's argument, 8 is the base of the logarithm, and 2 is the result of the logarithm.

Key Concepts

Exponential Equation

Exponential equations are mathematical expressions where a number, called the base, is raised to a certain power, which is the exponent. In simpler terms, it's a way to express how many times you multiply the base by itself. For example, in the equation \( 8^2 = 64 \), 8 is the base and 2 is the exponent. This means you multiply 8 by itself once, resulting in 64. Exponential equations are prevalent in various fields, from science to economics, as they can model growth processes, such as population growth or compound interest.
Exponential equations have the general form \( b^x = y \), where \( b \) is the base, \( x \) is the exponent, and \( y \) is the result. Understanding the structure of exponential equations is crucial for manipulating and converting them into different forms, such as logarithmic form.

Logarithms

Logarithms are the inverse operations of exponentiation, a bit like how division is the inverse of multiplication. They help us work out the exponent that needs to be applied to a base to get a certain number. For example, if you know that \( 8^2 = 64 \), a logarithm can help you figure out what power you need to raise 8 to get 64.
Logarithms are written as \( \log_b y = x \), where \( b \) is the base, \( y \) is the number you are taking the logarithm of, and \( x \) is the exponent or the power to which the base must be raised to produce \( y \). Using the exponential equation \( 8^2 = 64 \), the corresponding logarithmic form is \( \log_8 64 = 2 \). This tells you that 8 raised to the power of 2 gives you 64.

• The base of the logarithm must be a positive number other than 1.
• The argument of the logarithm, \( y \), must be a positive number.

These features make logarithms especially useful in solving exponential equations and understanding relationships involving exponential growth or decay.

Base and Exponent Identification

Identifying the base and the exponent in an exponential equation is the first crucial step when converting it into different forms, such as logarithmic. Recognizing these components helps to understand the relationship between the numbers involved. In the equation \( b^x = y \), \( b \) represents the base, \( x \) is the exponent or power, and \( y \) is the result.
For the equation \( 8^2 = 64 \), you can easily identify:

• Base \( (b) = 8 \)
• Exponent \( (x) = 2 \)
• Result \( (y) = 64 \)

Once you have identified these components, you can convert the equation to its logarithmic form: \( \log_8 64 = 2 \). Here, the base (8) and the result (64) become the base and the argument in the logarithmic expression, while the exponent (2) becomes the result of the logarithm. Understanding this conversion process is fundamental in moving between exponential and logarithmic forms, allowing for easier manipulation and solution of equations.