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Chapter 4: Problem 81

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. $$\log _{8} 0.59$$

Short Answer

Expert verified
\( \log_8{0.59} \approx -0.2536 \)

Step by step solution

01

Recall the Change-of-Base Theorem

The change-of-base theorem states that for any positive numbers \(a\), \(b\) and \(c\) (with \(a eq 1\) and \(b eq 1\)), the logarithm log base \(b\) of \(c\) can be rewritten using natural logarithms or common logarithms: \( \log_b{c} = \frac{ \log_a{c} }{ \log_a{b} } \).
02

Choose a Convenient Base

For simplicity, use base 10 (common logarithm). Thus, \( \log_8{0.59} = \frac{ \log_{10}{0.59} }{ \log_{10}{8} } \).
03

Calculate the Logarithms

Use a calculator to find the common logarithms: \( \log_{10}{0.59} \approx -0.2291 \) and \( \log_{10}{8} \approx 0.9031 \).
04

Divide the Logarithms

Divide the values obtained: \( \log_8{0.59} = \frac{ -0.2291 }{ 0.9031 } \approx -0.2536 \).

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Key Concepts

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

Logarithms
A logarithm answers the question: 'To what exponent must we raise a certain base to get a specific number?' For instance, in the equation \(\text{log}_2{8} = 3\), we ask 'To what power must we raise 2 to get 8?' The answer is 3, since \({2^3 = 8}\).

Logarithms have properties that make them very useful in mathematics:
  • Addition: \(\text{log}_b{(xy)} = \text{log}_b{x} + \text{log}_b{y}\)
  • Subtraction: \(\text{log}_b{(x/y)} = \text{log}_b{x} - \text{log}_b{y}\)
  • Exponent: \(\text{log}_b{(x^k)} = k \text{log}_b{x}\)

By understanding and using logarithms, we simplify solving complex mathematical problems, especially those involving exponential growth and decay. Logarithms appear frequently in science, engineering, and finance.
Base Conversion
Base conversion helps change the base of a logarithm to a more convenient one. The change-of-base theorem tells us how to do this efficiently:

\(\text{log}_b{c} = \frac{\text{log}_a{c}}{\text{log}_a{b}}\)

This formula allows us to convert a logarithm given in any base to another base. Here is a step-by-step way to use it:
  • Choose a new base, often 10 (common logarithm) or e (natural logarithm).
  • Calculate the logarithms in your new base for both the number and the original base.
  • Divide the results.

For example, to find \(\text{log}_8{0.59}\), we use base 10:
\(\text{log}_8{0.59} = \frac{\text{log}_{10}{0.59}}{\text{log}_{10}{8}}\).
This process simplifies dealing with logarithms of less familiar bases, leveraging more commonly utilized logarithms instead.
Common Logarithm
A common logarithm uses base 10 and is denoted as \(\text{log}_{10}\) or simply \(\text{log}\). Because of its base, it is used extensively in scientific contexts, especially when dealing with decibels or pH levels.
  • To calculate \(\text{log}_{10}{x}\), determine how many times 10 must be multiplied by itself to equal x.

For instance, \(\text{log}_{10}{1000} = 3\) because \({10^3 = 1000}\).

Electronic calculators almost always offer a 'log' button which simplifies direct calculation. To find \(\text{log}_8{0.59}\), we use the change-of-base theorem and calculate:

\(\text{log}_8{0.59} = \frac{\text{log}_{10}{0.59}}{\text{log}_{10}{8}}\).
Doing this, the approximate result is \({鈭�0.2536}\).
Natural Logarithm
A natural logarithm uses the base e (roughly 2.718) and is denoted as \(\text{log}_e\) or \(\text{ln}\). It is prevalent in mathematics, particularly in calculus.
  • To calculate \(\text{ln}(x)\), determine how many times e must be multiplied by itself to equal x.

For instance, \(\text{ln}(e^2) = 2\) because e raised to the power of 2 equals e^2.

Using natural logarithms can be helpful in simplifying exponentiation problems, slightly shifting the conversion formula:

\(\text{log}_b{c} = \frac{\text{ln}(c)}{\text{ln}(b)}\). For example, to find \(\text{log}_8{0.59}\), calculate:

\(\text{log}_8{0.59} = \frac{\text{ln}(0.59)}{\text{ln}(8)}}\).
Use this method when working within contexts like continuous compound interest or natural growth phenomena.

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Most popular questions from this chapter

For individual or collaborative investigation (Exercises \(117-122\) ) Assume \(f(x)=a^{x}\), where \(a>1 .\) Work these exercises in order. If \(\left.a=e, \text { what is the equation for } y=f^{-1}(x) ? \text { (You need not solve for } y .\right)\)

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{1}{x}, \quad x \neq 0$$

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Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts \((a)-(c)\) $$\begin{aligned} &\text { Given } f(x)=\ln x, \text { find }\\\ &\text { (a) } f\left(e^{6}\right)\\\ &\text { (b) } f\left(e^{\ln 3}\right) \quad \text { (c) } f\left(e^{2 \ln 3}\right) \end{aligned}$$
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