/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Write in exponential form. \(\lo... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 23

Write in exponential form. \(\log _{9} 3=\frac{1}{2}\)

Short Answer

Expert verified
3 = 9^{\frac{1}{2}}

Step by step solution

01

Identify the logarithmic equation

The given equation is \(\text{log}_{9} 3 = \frac{1}{2}\).
02

Recall the definition of logarithms

The equation \(\text{log}_{b} a = c\) can be rewritten in exponential form as \(a = b^c\).
03

Apply the definition

Using the definition, rewrite \(\text{log}_{9} 3 = \frac{1}{2}\) as \(3 = 9^{\frac{1}{2}}\).
04

Simplify the exponential expression

Recognize that raising 9 to the power of \(\frac{1}{2}\) is the same as taking the square root of 9. So, \(9^{\frac{1}{2}} = \sqrt{9} = 3\).

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Key Concepts

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

Logarithmic Equations
A logarithmic equation involves solving for a variable within a logarithm. In our exercise, we start with the logarithmic equation \(\text{log}_{9} 3 = \frac{1}{2}\). To understand this, we need to convert it into a more familiar format: an exponential form. Logarithmic equations can often look confusing, but breaking them down into steps helps. First, identify the base (here it鈥檚 9), the argument (here it鈥檚 3), and the result (here it鈥檚 1/2). We鈥檒l use these to rewrite the logarithmic equation into exponential form.
Definition of Logarithms
The second step of the exercise involves understanding the definition of logarithms. Logarithms are the inverse operations of exponents. The definition states that if \(\text{log}_{b} a = c\), then it can be rewritten in its exponential form as \(a = b^c\). This definition helps us convert the logarithmic form to exponential form. Applying this definition to the given equation \(\text{log}_{9} 3 = \frac{1}{2}\), we can rewrite it as \(3 = 9^{\frac{1}{2}}\). This crucial step makes it easier to solve and understand the relationship between the numbers involved.
Exponential Expressions
Finally, simplify the exponential expression. In our equation, \(9^{\frac{1}{2}}\) means raising 9 to the power of 1/2. An important thing to remember is that raising a number to the power of 1/2 is the same as taking its square root. Thus, \(9^{\frac{1}{2}} = \sqrt{9}\) and \sqrt{9} = 3\. This shows us that our logarithmic and exponential forms are equivalent: the logarithmic form \(\text{log}_{9} 3 = \frac{1}{2}\) is just another way of stating the exponential form \(3 = 9^{\frac{1}{2}}\). Understanding these conversions between forms is essential in algebra and further mathematics.

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Most popular questions from this chapter

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} 4$$

Solve each equation. \(\log _{x} 1=0\)

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{\sqrt[3]{x} \cdot \sqrt[5]{y}}{r^{2}}$$

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\log _{10}(x+3)+\log _{10}(x-3)$$

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{8}(9 \cdot 11)$$
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