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Chapter 4: Problem 38

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(x-1)+\log _{2}(x+1)=3$$

Short Answer

Expert verified
The solution to the equation is \(x = 3\).

Step by step solution

01

Combine the Logs

Use the property of logarithms that states \(\log_a(x) + \log_a(y) = \log_a(xy)\) to combine the two logarithm expressions on the left side of the equation. This yields \(\log_2((x-1)(x+1))=3\)
02

Convert to exponential form

The logarithmic equation can be expressed in exponential form as \(2^3 = (x-1)(x+1)\)
03

Solve for x

Now we can simply solve for x. First, \(2^3\) equals \(8.\) So, the equation simplifies to \(8=(x-1)(x+1).\) This expands to \(8 = x^2 - 1\). Adding \(1\) to both sides gives \(x^2 = 9\), so \(x = 卤3.\)
04

Check solutions

We need to make sure the solutions don't violate the logarithmic expressions' domain. Inserting \(x = 3\) into the original equation, both logarithms are defined, so \(3\) is a solution. Inserting \(x = -3\) into the original equation, the term \(\log_2 (x-1)\) isn't defined, so \(-3\) must be rejected.

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Most popular questions from this chapter

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

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. $$ 3 \ln x-\frac{1}{3} \ln y $$

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. $$ 3 \ln x+5 \ln y-6 \ln z $$

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. $$ \log _{b}\left(\frac{V x y^{3}}{z^{3}}\right) $$

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. $$ \log _{7}\left(\frac{7}{x}\right) $$
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