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Chapter 8: Problem 10

Write each equation in logarithmic form. $$ 8^{2}=64 $$

Short Answer

Expert verified
The logarithmic form of the equation \(8^2 = 64\) is \(log_8 64 = 2\).

Step by step solution

01

Understanding Exponential Form

In an exponential equation such as \(8^2 = 64\), the number 8 is the base, 2 is the exponent or power and 64 is the result. In other words, 8 raised to the power of 2 equals 64.
02

Conversion to Logarithmic Form

To transform this into logarithmic form, it should be read as 'log base 8 of 64 equals 2' or written as \(log_8 64 = 2\). This is the logarithmic form of the given equation.
03

Verifying the Solution

It's also possible to verify the result to ensure the change of form was done correctly. In this case, raising the base (8) to the power of the right-hand side of the new equation (2) should equal 64, which indeed it does.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Equations
An exponential equation is a mathematical expression where a base is raised to a power, known as the exponent. In simple terms, it's written in the form \(b^n = a\), where \(b\) is the base, \(n\) is the exponent, and \(a\) is the result. For example, in the equation \(8^2 = 64\), 8 is the base, 2 is the exponent, and 64 is the resulting value. Exponential equations are used to model scenarios involving growth, such as population increase, or decay, such as radioactive decay.
  • The base is the number being multiplied.
  • The exponent indicates how many times the base is multiplied by itself.
  • The result is what we get after carrying out the multiplication.
Recognizing exponential equations is the first step in converting them to other forms like logarithmic form.
Base and Exponent
The concept of base and exponent is crucial in understanding exponential equations. The base, often noted as \(b\), is the number that gets multiplied repeatedly. The exponent, written as \(n\), tells us how many times we multiply the base by itself.To illustrate, in the expression \(8^2\), 8 is the base. It means 8 is multiplied by itself once: \[8 \times 8 = 64\]The number 2 is the exponent showing this repeated multiplication. Higher exponents represent faster growth rates in exponential functions. Understanding how to calculate the value of an expression involving base and exponent helps in solving equations and converting them into different mathematical forms.
Logarithmic Form
When we want to express an exponential equation in a different form, we use logarithms. The logarithmic form helps us find the exponent required for a particular base to reach a given number.To convert the exponential equation \(8^2 = 64\) from exponential to logarithmic form, we are essentially asking: "What power do we raise 8 to, to get 64?" The answer is written as:\[\log_8 64 = 2\]Here, \(\log\) is the logarithm, 8 is the base, 64 is the number, and 2 is the exponent.
  • "\(\log_b a = n\)" implies that \(b^n = a\).
  • This form is useful for solving for missing exponents or simplifying complex multiplication problems.
  • Understanding logarithms is key in solving exponential equations and is widely used in science and engineering.

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