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Chapter 10: Problem 46

Solve each equation. $$\log _{10} x=-2$$

Short Answer

Expert verified
x = \(\frac{1}{100}\).

Step by step solution

01

Understand the Logarithmic Equation

Recognize that the given equation is a logarithmic equation of the form \(\text {log}_{10} x = -2\). This implies that if we can convert the logarithmic form to an exponential form, it will be easier to solve.
02

Convert Logarithmic Form to Exponential Form

To solve \(\text {log}_{10} x = -2\), rewrite it in its exponential form. The equivalent exponential form is \(10^{-2} = x\).
03

Solve for x

Calculate the value of \(10^{-2}\). This means calculating \(10^{-2}\), which equals \(\frac{1}{100}\). Therefore, \x = \frac{1}{100}\.

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Key Concepts

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

logarithmic to exponential form conversion
Logarithmic and exponential forms are two ways of expressing the same relationship. Understanding this conversion is crucial for solving logarithmic equations. The general form of a logarithmic equation is \(\text{log}_b(y) = x\). To convert this into its exponential form, you rewrite it as \[b^x = y.\] This makes solving for the variable easier in many cases.
solving logarithmic equations
When solving logarithmic equations, the first thing to consider is converting the logarithm into an exponential form. Take the given problem, \(\text{log}_{10} x = -2\). By converting this into an exponential form, you rewrite it as \[10^{-2} = x.\] Once in exponential form, you can directly solve for the variable. Always remember to check the base of the logarithm鈥攊t should match the base in the exponential form.
exponential equations
Exponential equations involve expressions where the variable is in the exponent. The previous logarithmic equation, converted to \10^{-2} = x\, is a simple exponential equation. To solve it, compute the power: \[10^{-2} = \frac{1}{10^2} = \frac{1}{100}.\] Therefore, \x = \frac{1}{100}.\ Simple exponential equations are usually solved by direct computation or transformation into a simpler base or exponent.

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Most popular questions from this chapter

Solve each equation. Give exact solutions. $$ \log _{6}(4 x+2)=2 $$

Solve each equation. Approximate solutions to three decimal places. $$ 6^{-x+1}=22 $$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{0.012 x}=23 $$

Solve each equation. Give exact solutions. $$ \log _{5}(12 x-8)=3 $$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{7} 4 $$
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