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Chapter 11: Problem 26

Evaluate expression. \(\log _{4} 4^{2}\)

Short Answer

Expert verified
The expression \( \log _{4} 4^{2} \) evaluates to 2.

Step by step solution

01

Understanding the Logarithm Identity

The expression given is \( \log_{4} 4^{2} \). Using the logarithmic identity \( \log_b (b^x) = x \), where \( b \) is the base and \( x \) is the exponent, we can simplify the expression. In this case, the base \( b \) is 4, and \( x \) is 2.
02

Applying the Logarithm Identity

Substitute the values from the expression into the logarithm identity. By applying \( \log_{4} (4^{2}) = 2 \), since the base of the logarithm \( 4 \) and the base of the exponent \( 4 \) are the same, the identity tells us that the expression equals the exponent 2.

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Key Concepts

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

Logarithmic Identity
Logarithmic identities help us simplify expressions by recognizing common patterns. In mathematics, one powerful identity is the logarithmic identity. This is particularly useful when the base of the logarithm and the base of the exponent match. For example, the identity \( \log_b (b^x) = x \) states that if you have a logarithm \( \log_b \) and it is followed by its base raised to some exponent \( x \), the output is simply \( x \). This identity works because logarithms and exponents are inverse operations, much like addition and subtraction or multiplication and division.
  • Use it to quickly simplify expressions where the base of the logarithm matches the base of the exponent.
  • Notice that this identity doesn鈥檛 depend on the magnitude of \( b \); it only matters that both bases match.
Understanding this identity streamlines calculations and makes solving such expressions much easier.
Exponent
An exponent indicates how many times a number, known as the base, is multiplied by itself. In mathematical terms, when you see a number in the form of \( b^x \), "\( b \)" is the base and "\( x \) " is the exponent. The notation can be broken down to mean: multiply \( b \) by itself \( x \) number of times. The role of the exponent in our exercise is crucial, particularly because it appears in a logarithmic expression \( \log_{4} 4^2 \). In this context, the identity used relies upon the exponent's relationship with the base.
  • Exponents can be thought of as repeated multiplication.
  • Any number raised to the zero power is 1. For example: \( 4^0 = 1 \).
  • An exponent of 1 means the number remains unchanged: \( b^1 = b \).
Understanding exponents is essential for breaking down and evaluating expressions, especially when applying logarithmic identities.
Base in Logarithms
The base in a logarithm is the number that gets raised to a power (the exponent) in the expression being evaluated. It's a foundational part of both the logarithm and the exponent expressions involved. In our problem \( \log_{4} 4^2 \), the number 4 serves as the base in both parts of the expression, making it easier to apply logarithmic identities.
  • The base of the logarithm must be positive and not equal to 1.
  • Changes in the base affect the evaluation of the logarithmic expression significantly.
  • When a logarithmic expression has the same base in the numerical part and the logarithmic part, simplification becomes straightforward using identities.
Understanding the base is pivotal for interpreting logarithmic expressions, as it ultimately determines how the expression is simplified or evaluated.

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Most popular questions from this chapter

Solve each equation. Give the exact solution and an approximation to four decimal places. See Example 5 . $$ e^{2.9 x}=4.5 $$

Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$\ln (-10)$$

Solve each equation. Give the exact solution and an approximation to four decimal places. See Example 5 . $$ e^{-0.7 x}=6.2 $$

Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest tenth. See Using Your Calculator: Solving Exponential Equations Graphically or Solving Logarithmic Equations Graphically. $$ 3^{x}-10=3^{-x} $$

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log 3 x=\log 9 $$
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