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Chapter 10: Problem 34

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

Short Answer

Expert verified
x=\frac{17}{2}

Step by step solution

01

Understand the given logarithmic equation

The given equation is \( \log_{6}(4x+2) = 2 \). This means that the logarithm base 6 of the expression \(4x + 2\) is equal to 2.
02

Convert logarithmic form to exponential form

To solve for \(x\), first convert the logarithmic equation to its exponential form. Recall that \( \log_{b}(a) = c \) is equivalent to \( b^{c} = a \). Thus, the equation \( \log_{6}(4x+2) = 2 \) can be written as: \[ 6^{2} = 4x + 2 \].
03

Simplify the exponential equation

Calculate \(6^{2}\) to get \(36\), so the equation becomes: \[ 36 = 4x + 2. \]
04

Solve for x

To isolate \(x\), subtract 2 from both sides: \[ 36 - 2 = 4x \], which simplifies to: \[ 34 = 4x. \] Then, divide both sides by 4: \[ x = \frac{34}{4} \]. Simplify the fraction to obtain: \[ x = \frac{17}{2} \].

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Key Concepts

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

solve logarithmic equations
Solving logarithmic equations involves finding the value of the variable that makes the equation true. In our example, we were given the logarithmic equation \(\log_{6}(4x+2) = 2\). The main steps to follow are:
  • Identify the logarithmic equation and understand the given logarithmic expression
  • Convert the logarithmic form to exponential form
  • Simplify the resulting equation
  • Finally, solve for the variable
The critical part here is to thoroughly understand each step and practice applying these steps to various logarithmic equations. This process allows us to break down the problem into manageable parts.
convert logarithmic form to exponential form
Converting a logarithmic equation to its exponential form is a crucial step in solving these kinds of problems. The general rule is that if you have \(\log_{b}(a) = c\), this can be converted to the exponential form \((b^c = a)\).
For our given equation \(\log_{6}(4x+2) = 2\), using this rule: the base 6 raised to the power of 2 equals \(4x + 2\), we get \(6^2 = 4x + 2\).
Why is this helpful? Because converting to exponential form often simplifies the process of solving for the variable. Now, instead of working with a logarithmic expression, we work with a straightforward exponential equation which is simpler to solve.
simplifying exponential equations
After converting to an exponential form, we often get a simpler equation that we can solve using basic algebra. For the example \(6^2 = 4x + 2\), simplifying it involves:
  • Calculating \((6^2)\) which gives 36
  • Writing the equation as \(36 = 4x + 2\)
  • Isolating \(x\) by moving 2 to the left side:
  • \(36 - 2 = 4x\)
  • This simplifies to \(34 = 4x\)
  • Finally, divide both sides by 4, resulting in \(x = \frac{34}{4}\) and further simplifying to \(x = \frac{17}{2}\)
Breaking down the exponential equation and solving it step-by-step can greatly help in understanding the process. Each part of the equation can be gradually simplified to isolate and find the value of the variable.

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Most popular questions from this chapter

Graph each logarithmic function. $$f(x)=\log _{1 / 5} x$$

Solve each equation. Give exact solutions. $$ \log _{2} x+\log _{2}(x+4)=5 $$

An online sales company finds that its sales (in millions of dollars) are approximated by the logarithmic function $$S(x)=\log _{2}(3 x+1)$$ where \(x\) is the number of advertisements placed on a popular website. How many advertisements must be placed to earn sales of \(\$ 4\) million?

Solve each equation. Give exact solutions. $$ \log _{2}(2 x-1)=5 $$

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