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Chapter 9: Problem 48

Solve each equation. Check your solutions. \(\log _{b} 64=3\)

Short Answer

Expert verified
The base \( b \) is 4.

Step by step solution

01

Understand the Logarithmic Equation

The equation given is \( \log_{b} 64 = 3 \). This means that when base \( b \) is raised to the power of \( 3 \), we should get the value 64.
02

Convert Logarithmic Form to Exponential Form

The given logarithmic equation can be rewritten in an exponential form as follows: \[b^3 = 64\]. This step allows us to solve for the base \( b \).
03

Solve for the Base

We need to find the value of \( b \) such that the equation \( b^3 = 64 \) holds true. Since \( 4^3 = 64 \), the base \( b \) must be 4. Thus, \( b = 4 \).
04

Verify the Solution

Now, we need to check that the solution \( b = 4 \) satisfies the original logarithmic equation: \( \log_{4} 64 = 3 \).Since 64 is indeed \( 4^3 \), the solution is correct.

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Key Concepts

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

Exponential Form Conversion
When dealing with logarithmic equations, a useful skill to have is the ability to convert them into exponential form. The equation \( \log_{b} 64 = 3 \) in logarithmic terms tells us that the base \( b \) raised to the exponent 3 equals 64. To convert this to exponential form, we rearrange this information to write it as \( b^3 = 64 \). This conversion is often straightforward but crucial as it simplifies the problem into a more familiar form. Here's why it's so beneficial:
  • It reveals the direct relationship between the base, the exponent, and the result.
  • It turns a potentially abstract logarithmic statement into something more intuitive and tangible.
By converting to exponential form, you set the stage for standard algebraic techniques, allowing you to solve for \( b \) with ease.
Base Solving
Now that the equation is in exponential form \( b^3 = 64 \), the task is to find the value of \( b \). To solve for \( b \):- Recognize common powers: Here, 64 is a common power, specifically \( 4^3 \), meaning that the base \( b \) must be 4.- Utilize familiar exponent rules: If you already know that \( 4^3 = 64 \), you can quickly deduce that \( b = 4 \).Recognizing familiar powers or using a calculator to find the cube root of 64 will both yield \( b = 4 \). This step reverses the initial exponential statement and isolates the base, revealing it as a specific number, 4. It's essential to approach this step logically, keeping in mind the meaning of powers and roots.
Solution Verification
After determining \( b = 4 \), it鈥檚 vital to verify the solution to ensure accuracy. We must confirm that this base satisfies the original logarithmic equation. To verify:- Substitute \( b = 4 \) back into the original logarithmic form, \( \log_{4} 64 = 3 \).- Check that indeed \( 4^3 = 64 \), which aligns with the earlier exponential form.Verification is a crucial part of problem-solving. It ensures that no misstep has occurred along the way. By confirming that the original condition holds when \( b = 4 \), you affirm the correctness of your entire solution process, granting confidence in your answer.

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