Problem 61 Solve the logarithmic equation a... [FREE SOLUTION]

Chapter 3: Problem 61

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$$

Short Answer

Expert verified

The solution to the equation $\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$ is $x = 2$.

Step by step solution

Combine Logs

Use the properties of logarithms to combine $\log _{4} x-\log _{4}(x-1)$ into a single logarithm. According to the properties of logarithms, $\log_b(a) - \log_b(c) = \log_b(a/c)$, thus, the equation becomes $\log_4\left(\frac{x}{x-1}\right) = \frac{1}{2}$

Convert to Exponential Form

Consider that the logarithm equation $\log_b(a) = c$ is equivalent to the exponential equation $b^c = a$. Therefore, we can re-write our equation in its exponential equivalent form as follows: $4^{0.5} = \frac{x}{x-1}$

Solve for $x$

Next, solve this equation for $x$. First, simplify $4^{0.5}$ which gives $2$, then, cross-multiply and rearrange terms to isolate $x$. \[2 = \frac{x}{x-1} \Rightarrow 2x - 2 = x \Rightarrow x = 2\]

Validate Solution

Finally, check if the solution $x = 2$ is valid by replacing $x$ with $2$ in the original equation, and confirm that both sides are equal. \[\log _{4} 2-\log _{4}(2-1)=\frac{1}{2} \Rightarrow \frac{1}{2} - 0 = \frac{1}{2}\] The solution is valid.

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91影视

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$$

Short Answer

Step by step solution

Combine Logs

Convert to Exponential Form

Solve for \(x\)

Validate Solution

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