/*! 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 65 Solve equation. \(\log _{3}(x+... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 65

Solve equation. \(\log _{3}(x+4)=2\)

Short Answer

Expert verified
x = 5

Step by step solution

01

- Understand the Logarithmic Equation

The equation given is \(\log _{3}(x+4)=2\). This is a logarithmic equation where the base of the logarithm is 3.
02

- Rewrite the Logarithmic Equation in Exponential Form

Recall that \[ \log_{b}(a) = c \] can be rewritten as \[ b^{c} = a \]. Applying this to the given equation, \[3^{2} = x + 4\].
03

- Solve for x

Simplify the exponential equation to find the value of x. We have \[ 3^{2} = 9 \], so the equation becomes \[ 9 = x + 4 \]. Subtract 4 from both sides to isolate x: \[ x = 5 \].
04

- Verify the Solution

Substitute \x = 5\ back into the original logarithmic equation to ensure it is correct: \[ \log_{3}(5 + 4) = \log_{3}(9) = 2 \]. This is true, confirming the solution.

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Key Concepts

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

Logarithmic Functions
Logarithmic functions are used to find the exponent that a base must be raised to produce a given number. For example, the function \( \log_{3}(x+4) \) in the given equation helps us understand what power 3 must be raised to get \( x+4 \). The value of this logarithmic function when it is equal to 2 can be written as an exponential form to solve for x.
Exponential Form
To solve a logarithmic equation, one common technique is to rewrite it in its exponential form. The relationship between logarithmic and exponential forms is key. We know that \[ \log_{b}(a) = c \] can be rewritten accurately as \[ b^c = a \].
In our exercise, we start with the logarithmic form \( \log_{3}(x+4)=2 \). Rewriting this using the said relationship, we convert it into \( \3^{2} = x + 4 \), making it easier to isolate and solve for x. This principle can be applied to any logarithm equation to make it solvable.
Solving Equations
Once the logarithmic equation is rewritten in its exponential form, solving for the variable becomes straightforward.
For our exercise, we now work with \[ 3^{2} = x + 4 \]. Simplifying this, we get \3^{2} = 9 \. Hence, the new equation becomes \[ 9 = x + 4 \]. To isolate x, we subtract 4 from both sides, resulting in \[ x = 5 \].
Always verify your solution by substituting it back into the original equation. Plugging \( x = 5 \) back in, we get \[ \log_{3}(5+4) = \log_{3}(9) = 2 \], matching our original equation. This confirms that \( x = 5 \) is indeed the correct solution.

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Most popular questions from this chapter

Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window. \(f(x)=x^{3}+5\)

Let \(k\) represent the number of letters in your last name. (a) Use your calculator to find \(\log k\) (b) Raise 10 to the power indicated by the number in part (a). What is your result? (c) Use the concepts of Section 12.1 to explain why you obtained the answer you found in part (b). Would it matter what number you used for \(k\) to observe the same result?

Without using a calculator, give the value of \(\log 10^{31.6}\)

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