Sure, let's solve the equation step by step:
[tex]\[
\log _{10}\left(4 x^2+1\right)-2 \log _{10} x-\log _{10} 2=1
\][/tex]
### Step 1: Apply properties of logarithms
First, we combine the logarithmic terms using logarithm properties. Recall that [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex]. We can combine the terms as follows:
[tex]\[
\log _{10}\left(4 x^2+1\right)-2 \log _{10} x = \log _{10}\left(4 x^2+1\right)-\log _{10}(x^2)^2 = \log _{10}\left(\frac{4 x^2+1}{x^2}\right)
\][/tex]
Next, we bring [tex]\(-\log_{10} 2\)[/tex] into the logarithm:
[tex]\[
\log_{10}\left(\frac{4x^2 + 1}{x^2}\right) - \log_{10}(2) = \log_{10}\left(\frac{4x^2 + 1}{2x^2}\right)
\][/tex]
So the equation becomes:
[tex]\[
\log_{10}\left(\frac{4x^2 + 1}{2x^2}\right) = 1
\][/tex]
### Step 2: Solve the logarithmic equation
Since [tex]\(\log_{10}(a) = b\)[/tex] implies [tex]\(a = 10^b\)[/tex], we convert the logarithmic equation to its exponential form:
[tex]\[
\frac{4x^2 + 1}{2x^2} = 10^1
\][/tex]
So we get:
[tex]\[
\frac{4x^2 + 1}{2x^2} = 10
\][/tex]
### Step 3: Simplify and solve the resulting algebraic equation
Multiply both sides of the equation by [tex]\(2x^2\)[/tex] to clear the fraction:
[tex]\[
4x^2 + 1 = 20x^2
\][/tex]
Move all terms to one side to form a standard quadratic equation:
[tex]\[
4x^2 + 1 - 20x^2 = 0
\][/tex]
Combine like terms:
[tex]\[
-16x^2 + 1 = 0
\][/tex]
Isolate [tex]\(x^2\)[/tex]:
[tex]\[
-16x^2 = -1
\][/tex]
Divide both sides by [tex]\(-16\)[/tex]:
[tex]\[
x^2 = \frac{1}{16}
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \frac{1}{4}
\][/tex]
### Step 4: Verify the solution
To verify the solution, substitute [tex]\(x = \frac{1}{4}\)[/tex] and [tex]\(x = -\frac{1}{4}\)[/tex] back into the original equation to check for valid solutions.
However, since the logarithm function is not defined for negative arguments in this context (as both [tex]\(\log_{10} x\)[/tex] and the argument of the logarithm must be positive), we discard the negative solution.
Thus, we conclude that the valid solution is:
[tex]\[
\boxed{\frac{1}{4}}
\][/tex]
