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Chapter 4: Problem 22

Evaluate each expression without using a calculator. $$\log _{7} 49$$

Short Answer

Expert verified
The value of \(\log_{7}{49}\) is 2.

Step by step solution

01

Understand the logarithmic notation

A logarithm \(\log_{b}{a}\) is the exponent to which we must raise the base \(b\) to get the number \(a\). So if you have \(\log_{b}{a}=x\), then you should think of it as \(b^x=a\). You can apply this concept to our problem \(\log_{7}{49}\). The problem is actually asking what power you have to raise 7 (the base) in order to get 49.
02

Apply the power of base to the logarithmic expression

In this case, you need to think about what power to raise 7 in order to get 49. In this instance, you'll realize that if you square 7 (i.e., \(7^2=49\)), you get 49. Hence, the power is 2.
03

Write the final result

So, according to the calculation and the notation of the logarithm, \(\log_{7}{49}\) equals 2.

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Most popular questions from this chapter

Describe the power rule for logarithms and give an example.

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{6} 17 $$

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 4 \ln x+7 \ln y-3 \ln z $$

If \(\ S 4000\) is deposited into an account paying \(3 \%\) interest compounded annually and at the same time \(\ S 2000\) is deposited into an account paying \(5 \%\) interest compounded annually, after how long will the two accounts have the same balance?
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