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Chapter 12: Problem 11

Write in logarithmic form. \(\sqrt[4]{625}=5\)

Short Answer

Expert verified
\( \log_{625}(5)=\frac{1}{4} \)

Step by step solution

01

Understand the Exercise

The task is to write the given expression in logarithmic form. The given expression is \(\sqrt[4]{625}=5\).
02

Rewrite the Expression in Exponential Form

Before converting to logarithmic form, rewrite the equation \( \sqrt[4]{625}=5 \) in its equivalent exponential form. \( \sqrt[4]{625}=625^{1/4}=5 \)
03

Set Up Logarithmic Form

Recall that the logarithmic form of \( a^b = c \) is \( \log_a(c) = b \). For the expression \( 625^{1/4}=5 \), identify the base \( a \), the exponent \( b \), and the result \( c \). Here, the base \( a = 625 \), the exponent \( b = 1/4 \), and the result \( c = 5 \).
04

Convert to Logarithmic Form

Apply the logarithmic form. The equivalent logarithmic form of the equation \( 625^{1/4}=5 \) is \( \log_{625}{5}=\frac{1}{4} \).

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Key Concepts

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

exponential form
To understand logs, you need to first understand exponentials. The exponential form involves expressions like: \(a^b = c\), where:
  • The base (\(a\)) is the number being multiplied repeatedly.
  • The exponent (\b) tells you how many times to multiply the base.
Here鈥檚 a simple example: \(2^3 = 8\). This means 2 is multiplied by itself 3 times: \(2 \times 2 \times 2 = 8\). In our exercise: \(\root[4]{625}=5\) is written as: \(625^{1/4}=5\). It states that raising 625 to the power of 1/4 gives 5.
logarithmic conversion
Logs are another way to express exponentials. The log form of \(a^b = c\) is \(\text{log}_a(c) = b\). It helps us go from the result back to the exponent. For example: if \(2^3 = 8\), then \(\text{log}_2(8) = 3\). For our specific problem:
Given \(625^{1/4}=5\), we convert it to \( \text{log}_{625}(5) = 1/4\). This tells us that 625 raised to the power of 1/4 equals 5.
base and exponent identification
Identifying the base and exponent in a logarithmic or exponential expression is key.
Here is how you can determine them:
  • **Base (\footnotesize{a})**: The number being repeated/raised to a power.
  • **Exponent (\footnotesize{b})**: The power we raise the base to.
  • The **result (\footnotesize{c})**: The final number after raising the base to exponent.
In the problem:

\root[4]{625}=5 -> \text{equivalent to} \( 625^{1/4}=5\), the base (a) is 625, the exponent (b) is 1/4, and result (c) is 5. Converting this to log: \text{log}_{625}(5)=1/4.

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Most popular questions from this chapter

Solve each equation. \(\log _{x} 9=\frac{1}{2}\)

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Determine whether each statement is true or false. $$\log _{3} 49+\log _{3} 49^{-1}=0$$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{\pi} 10\)

Solve each equation. \(\log _{4} \sqrt{64}=x\)
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