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Chapter 4: Problem 90

Explain why the logarithm of 1 with base \(b\) is 0.

Short Answer

Expert verified
The logarithm of 1 with any base \(b\) is 0, denoted as \(\log_b(1) = 0\), as according to the properties of exponentiation any number raised to the power 0 gives 1.

Step by step solution

01

Understand the logarithm function

The logarithm function is the inverse of the exponential function. Given a logarithm function with base \(b\) and argument \(x\), denoted as \(\log_b(x)\), it represents the exponent to which the base \(b\) must be raised to get \(x\). In other words, if we have \(b^{y} = x\), then \(\log_b(x) = y\), where \(b > 0\), \(b \neq 1\), and \(x > 0\).
02

Apply the logarithm property

We can apply this property to the problem at hand. We are asked to find \(\log_b(1)\) for an arbitrary base \(b\). So we need to find some number \(y\) such that raising \(b\) to the power of \(y\) gives us \(1\).
03

Reason about the exponent needed

It's a common mathematical fact that any number to the power of 0 is 1. That is, \(b^{0} = 1\) for any number \(b\). This is because any number multiplied by itself zero times results in 1. So, to get an argument of 1 for our logarithm function, we need an exponent of 0.
04

Input into the logarithm function

So based on Step 3, we can say that \(\log_b(1) = 0\). This is because \(b^{0} = 1\), which is a property of exponentiation. Therefore any logarithm having an argument of 1 is 0, no matter what the base is.

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Most popular questions from this chapter

You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; 1-\frac{1}{2}+\frac{1}{3} ; 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} ; \ldots .\) Describe what you observe.

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b} x^{7} $$

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$

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. $$ 8 \ln (x+9)-4 \ln x $$
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