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Chapter 5: Problem 69

Find the logarithm using common logarithms and the change-of-base formula. $$\log _{4} 100$$

Short Answer

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Step by step solution

01

Understand the Logarithm Problem

To find \(\log_{4} 100\text{\)}, we want to determine the exponent to which 4 must be raised to get 100.
02

Apply the Change-of-Base Formula

The change-of-base formula states \(\log_{a} b = \frac{\log_{c} b}{\log_{c} a}\text{\)}. For common logarithms, we use base 10 (denoted as \(\log \text{\)}). Thus, \(\log_{4} 100 = \frac{\log 100}{\log 4}\text{\)}.
03

Calculate the Numerator

Find \(\log 100\text{\)}. Since \(\log 100 = 2\text{\)} (because 10^2 = 100), we have the numerator \(\log 100 = 2\text{\)}.
04

Calculate the Denominator

Find \(\log 4\text{\)}. Since \(\log 4 = \log (2^2) = 2 \log 2\text{\)}, we need the value of \(\log 2\text{\)}. Using common logarithm tables or a calculator, \(\log 2 \approx 0.301\text{\)}, so \(\log 4 = 2 \times 0.301 = 0.602\text{\)}.
05

Divide Numerator by Denominator

Using the values from Steps 3 and 4, compute \(\frac{2}{0.602} \approx 3.32\text{\)}.

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Key Concepts

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

Logarithms
Logarithms are the inverse operations to exponents. They help us solve questions like, 'to what power must a base be raised to get a certain number?' For example, \(\text{log}_{2} 8 \) asks to which power 2 must be raised to obtain 8. Since 2^3 = 8, \(\text{log}_{2} 8 = 3 \). Understanding logarithms is crucial for handling different kinds of equations in algebra and beyond.
Common Logarithms
Common logarithms are logarithms that use 10 as the base. They are often written simply as \(\text{log} \) without displaying the base. This type of logarithm is particularly useful because it relates directly to our base-10 number system, making it easier to perform calculations with. For instance, \(\text{log} 100 \) equals 2 because 10 squared equals 100 (10^2 = 100). \(\text{log} \) functions are commonly found on scientific calculators.
Exponents
Exponents provide a shorthand way of expressing repeated multiplication of a number by itself. For example, 4 raised to the power of 3 is written as 4^3 and equals 4 x 4 x 4, which is 64. They are fundamental in understanding logarithms. When we say \(\text{log}_{4} 100 \), we are asking: '4 raised to what power equals 100?' Applying the change-of-base formula helps us solve this. By converting it to common logarithms, we can effectively find the answer by dividing \(\text{\text{log}} 100 \) by \(\text{\text{log}} 4 \).

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Most popular questions from this chapter

Express as a single logarithm and, if possible, simplify. $$\log _{a}\left(x^{2}+x y+y^{2}\right)+\log _{a}(x-y)$$

Use a graphing calculator to find the approximate solutions of the equation. $$\log _{3} x+7=4-\log _{5} x$$

Compound Interest. Suppose that \(\$ 82.000\) is invested at \(4 \frac{1}{2} \%\) interest, compounded quarterly. a) Find the function for the amount to which the investment grows after \(t\) years. b) Graph the function. c) Find the amount of money in the account at \(t=0,2\) \(5,\) and 10 years. d) When will the amount of money in the account reach \(\$ 100,000 ?\)

Express as a sum or a difference of logarithms. $$\log _{a} \frac{x-y}{\sqrt{x^{2}-y^{2}}}$$

Use a graphing calculator to find the approximate solutions of the equation. $$5 e^{5 x}+10=3 x+40$$
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