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Chapter 10: Problem 331

In the following exercises, solve each equation. \(\log _{4}(7 x+15)=3\)

Short Answer

Expert verified
x = 7

Step by step solution

01

- Understand the Problem

The given equation is \(\text{log}_{4}(7x+15)=3\). The goal is to solve for the variable \(x\).
02

- Rewrite the Logarithmic Equation

Rewrite the equation in exponential form. The equation \( \text{log}_{4}(7x+15)=3\) can be rewritten as \( 4^3 = 7x + 15 \).
03

- Simplify the Exponential Equation

Evaluate the exponential expression on the left side of the equation: \( 4^3 = 64\). So the equation becomes \( 64 = 7x + 15 \).
04

- Isolate the Variable

Subtract 15 from both sides to isolate the term with \(x\): \( 64 - 15 = 7x \). This simplifies to \( 49 = 7x \).
05

- Solve for \(x\)

Divide both sides by 7 to solve for \(x\): \( x = \frac{49}{7} \). Therefore, \( x = 7 \).

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Key Concepts

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

logarithms
Logarithms are the inverse operations of exponentiation. They help us answer the question: 'To what power do we need to raise a base number to get another number?' For example, if we have \(\text{log}_{4}(7x+15)=3\), it means we need to raise 4 to the power of 3 to get 7x + 15.

Here are some useful properties of logarithms to remember:
  • \(\text{log}_{b}(b^n) = n \): The logarithm of a base raised to an exponent is simply the exponent.
  • \(\text{log}(AB) = \text{log}(A) + \text{log}(B)\): The logarithm of a product is the sum of the logarithms.
  • \(\text{log}\bigg(\frac{A}{B}\bigg) = \text{log}(A) - \text{log}(B)\): The logarithm of a quotient is the difference of the logarithms.
In the exercise, we used the first property to transform a logarithmic equation into an exponential equation

which makes it easier to solve for the variable.
exponential equations
An exponential equation is one in which a variable appears in the exponent. To solve such equations, you often convert logarithms to exponential form.

For instance, the equation from the problem \(\text{log}_{4}(7x+15)=3\) can be rewritten as \ 4^3 = 7x + 15 \. This step utilizes the definition of logarithms, linking the logarithmic and exponential forms.

Let's see this in action:
  • \ \text{log}_{b}(A) = C \ translates to \ b^C = A \.
  • In our example, \ \text{log}_{4}(7x+15)=3 \ becomes \ 4^3 = 7x + 15 \.
After simplifying the exponential, we move closer to isolating the variable.
isolating the variable
Isolating the variable is the process of solving for a variable by performing arithmetic operations to isolate it on one side of the equation.

In our problem, we have \ 4^3 = 7x + 15 \. First, evaluate the left side: \ 4^3 = 64 \. So, the equation becomes \ 64 = 7x + 15 \. To isolate \(x\), follow these steps:
  • Subtract 15 from both sides: \ 64 - 15 = 7x \ which simplifies to \ 49 = 7x \.
  • Divide both sides by 7: \ \frac{49}{7} = x \, so \ x = 7 \.
This step-by-step approach ensures that you correctly solve for the variable and understand the transformations happening to it. Understanding how to isolate the variable is key to solving any equation.

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Most popular questions from this chapter

In the following exercises, solve for \(x\). \(\log _{3}(2 x-1)=\log _{3}(x+3)+\log _{3} 3\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(\frac{1}{4} e^{x}=3\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{2} 5-\log _{2}(x-1)\)

In the following exercises, solve. Sung Lee invests \(\$ 5,000\) at age 18 . He hopes the investments will be worth \(\$ 10,000\) when he turns \(25 .\) If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?

In the following exercises, solve each equation. \(3 \log x=\log 125\)
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