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Chapter 10: Problem 201

In the following exercises, solve each logarithmic equation. \(\log _{4}(3 x-2)=2\)

Short Answer

Expert verified
The solution is \(x = 6\).

Step by step solution

01

Understand the Equation

The given equation is \(\text{log}_{4}(3x-2)=2\). The goal is to find the value of \(x\).
02

Convert the Logarithmic Equation to Exponential Form

Recall that \(\log_{b}(a) = c\ is equivalent to \ b^c = a\). Using this property, convert \(\log_{4}(3x-2)=2\) to the exponential form: \(4^2 = 3x-2\).
03

Simplify the Exponential Equation

Calculate \(4^2\): \(4^2 = 16\). Now the equation is \(16 = 3x - 2\).
04

Solve for x

Solve for \(x\) by isolating it on one side of the equation. Add 2 to both sides: \(16 + 2 = 3x\), which simplifies to \(18 = 3x\). Next, divide both sides by 3: \(x = \frac{18}{3}\), which simplifies to \(x = 6\).
05

Verify the Solution

Substitute \(x = 6\) back into the original equation to verify the solution: \(\log_{4}(3 \cdot 6 - 2) = \log_{4}(18 - 2) = \log_{4}(16) = 2\). Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

Exponential Form Conversion
To solve a logarithmic equation, a key step is often converting it into its exponential form. This helps simplify the equation and makes it easier to solve. When you see \(\text{log}_{b}(a) = c\), you can rewrite it in the exponential form as \(b^c = a\). This means you're using the base of the logarithm raised to the power of the result to get the original argument. For instance, in the equation \(\text{log}_{4}(3x-2)=2\), we convert it to the exponential form as \(4^2 = 3x-2\). This conversion is crucial because it transforms the logarithmic equation into a more familiar format that is easier to solve.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. Understanding their properties is essential for solving logarithmic equations. For example:
  • \( \text{log}_{b}(xy) = \text{log}_{b}(x) + \text{log}_{b}(y) \)
  • \( \text{log}_{b}(x/y) = \text{log}_{b}(x) - \text{log}_{b}(y) \)
  • \( \text{log}_{b}(x^y) = y\text{log}_{b}(x) \)
These properties help to break down complex logarithmic expressions into simpler components. In our example, \( \text{log}_{4}(3x-2)=2\), we don鈥檛 need these additional properties, but they are useful to know for more complex situations. The main idea here is to recognize that logarithms help us solve equations involving exponents by transforming multiplicative relationships into additive ones, and vice versa.
Solving Equations
Once we've converted the logarithmic equation to an exponential form, solving it is straightforward. Here鈥檚 a simple step-by-step strategy:
  • First, isolate the exponential term if it鈥檚 not already alone. For example, in \(4^2 = 3x-2\), the term is already isolated.
  • Next, compute the value of the exponential expression. Here, \(4^2 = 16\).
  • Then, solve the resulting linear equation. Add or subtract terms to isolate the variable. In our example: \(16 = 3x - 2\) translates to \(18 = 3x\) after adding 2 to both sides.
  • Finally, solve for the variable by dividing or multiplying as necessary. Here, divide both sides by 3: \(x = 6\).
Always double-check your solution by substituting it back into the original equation. Verifying helps ensure that the solution is correct. In our example, substituting \(x = 6\) back into \( \text{log}_{4}(3 \times 6 - 2) = 2\) confirms that \( \text{log}_{4}(16) = 2\), which holds true.

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Most popular questions from this chapter

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

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible. \(\log _{3}\left(\sqrt{2} x^{2}\right)\)

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log x^{-2}\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(e^{x}=16\)

In the following exercises, solve for \(x\). \(\log _{5}(x+3)+\log _{5}(x-6)=\log _{5} 10\)
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