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Chapter 13: Problem 31

Explain how to solve a logarithmic equation of the form \(\log _{a} b=c\)

Short Answer

Expert verified
To solve a logarithmic equation of the form \(\log_a b = c\), first convert it to its exponential form \(a^c = b\). Then, solve for the unknown variable by isolating it using algebraic techniques. For example, if given \(\log_2 x = 3\), convert it to \(2^3 = x\) and find the value of x as \(x = 8\).

Step by step solution

01

Understand the logarithmic equation

Given a logarithmic equation in the form: \(\log_a b = c\), where \(a\) is the base, \(b\) is the argument (the number we are taking the logarithm of), and \(c\) is the result. Our goal is to find the value of the variable in the equation.
02

Convert the logarithmic equation to exponential form

Using the definition of logarithms, we can convert the logarithmic equation to an exponential form: \(a^c = b\) Here, the base \(a\) raised to the power of the result \(c\) is equal to the argument \(b\).
03

Solve for the variable

Now that you have the equation in the exponential form, you can solve for the unknown variable depending on whether the base, argument, or result is unknown. If the variable is not already isolated, use algebraic techniques to isolate and find the value of the variable. For example, if the equation is \(\log_2 x = 3\), you can rewrite it in the exponential form as: \(2^3 = x\) Then, calculate the value of 2 raised to the power of 3: \(2^3 = 8\) So, the value of x is 8: \(x = 8\)

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Most popular questions from this chapter

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log _{7} d-\log _{7} 3$$

Find the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$g(x)=\sqrt[3]{x}+4$$

Find the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$h(x)=-\frac{1}{3} x$$

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log _{8} t+2 \log _{8} u-3 \log _{8} v$$

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log _{p} r-\log _{p} s$$
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