Answer :
To solve the equation [tex]\(\log_{x^2} 16 - \log_{\sqrt{x}} 7 = 2\)[/tex], we can break down the problem into simpler steps.
### Step 1: Rewrite the Logarithms
Firstly, let's understand what each part of the equation implies:
1. [tex]\(\log_{x^2} 16\)[/tex] means the power to which [tex]\(x^2\)[/tex] must be raised to get 16.
2. [tex]\(\log_{\sqrt{x}} 7\)[/tex] means the power to which [tex]\(\sqrt{x}\)[/tex] must be raised to get 7.
### Step 2: Convert Logarithms using Change of Base
We can use the change of base formula for logarithms:
[tex]\[
\log_b a = \frac{\log_k a}{\log_k b}
\][/tex]
This formula applies regardless of whether the base [tex]\(k\)[/tex] is the common logarithm (base 10) or the natural logarithm (base [tex]\(e\)[/tex]).
So,
[tex]\[
\log_{x^2} 16 = \frac{\log 16}{\log x^2}
\][/tex]
[tex]\[
\log_{\sqrt{x}} 7 = \frac{\log 7}{\log \sqrt{x}}
\][/tex]
### Step 3: Solve Each Logarithm
Using properties of logarithms for simplicity:
[tex]\[
\log x^2 = 2\log x
\][/tex]
[tex]\[
\log \sqrt{x} = \frac{1}{2}\log x
\][/tex]
Substituting these back into the equations:
[tex]\[
\log_{x^2} 16 = \frac{\log 16}{2\log x}
\][/tex]
[tex]\[
\log_{\sqrt{x}} 7 = \frac{\log 7}{\frac{1}{2}\log x} = \frac{2\log 7}{\log x}
\][/tex]
### Step 4: Combine and Simplify
Plug these into the original equation:
[tex]\[
\frac{\log 16}{2\log x} - \frac{2\log 7}{\log x} = 2
\][/tex]
To combine the fractions:
[tex]\[
\frac{\log 16 - 4\log 7}{2\log x} = 2
\][/tex]
### Step 5: Clear the Fraction and Solve for [tex]\(x\)[/tex]
Multiply both sides by [tex]\(2\log x\)[/tex] to clear the fraction:
[tex]\[
\log 16 - 4\log 7 = 4\log x
\][/tex]
Now solve for [tex]\(\log x\)[/tex]:
[tex]\[
4\log x = \log 16 - 4\log 7
\][/tex]
Divide by 4:
[tex]\[
\log x = \frac{\log 16 - 4\log 7}{4}
\][/tex]
### Step 6: Determine [tex]\(x\)[/tex]
Calculate the numerical values (or leave in terms of logs if calculation not done):
- [tex]\(\log 16 = 4 \log 2\)[/tex]
- Using the given result, the value of [tex]\(x\)[/tex] that satisfies this equation is [tex]\(x = \frac{2}{7}\)[/tex].
Thus, the solution to the equation is [tex]\(x = \frac{2}{7}\)[/tex].
### Step 1: Rewrite the Logarithms
Firstly, let's understand what each part of the equation implies:
1. [tex]\(\log_{x^2} 16\)[/tex] means the power to which [tex]\(x^2\)[/tex] must be raised to get 16.
2. [tex]\(\log_{\sqrt{x}} 7\)[/tex] means the power to which [tex]\(\sqrt{x}\)[/tex] must be raised to get 7.
### Step 2: Convert Logarithms using Change of Base
We can use the change of base formula for logarithms:
[tex]\[
\log_b a = \frac{\log_k a}{\log_k b}
\][/tex]
This formula applies regardless of whether the base [tex]\(k\)[/tex] is the common logarithm (base 10) or the natural logarithm (base [tex]\(e\)[/tex]).
So,
[tex]\[
\log_{x^2} 16 = \frac{\log 16}{\log x^2}
\][/tex]
[tex]\[
\log_{\sqrt{x}} 7 = \frac{\log 7}{\log \sqrt{x}}
\][/tex]
### Step 3: Solve Each Logarithm
Using properties of logarithms for simplicity:
[tex]\[
\log x^2 = 2\log x
\][/tex]
[tex]\[
\log \sqrt{x} = \frac{1}{2}\log x
\][/tex]
Substituting these back into the equations:
[tex]\[
\log_{x^2} 16 = \frac{\log 16}{2\log x}
\][/tex]
[tex]\[
\log_{\sqrt{x}} 7 = \frac{\log 7}{\frac{1}{2}\log x} = \frac{2\log 7}{\log x}
\][/tex]
### Step 4: Combine and Simplify
Plug these into the original equation:
[tex]\[
\frac{\log 16}{2\log x} - \frac{2\log 7}{\log x} = 2
\][/tex]
To combine the fractions:
[tex]\[
\frac{\log 16 - 4\log 7}{2\log x} = 2
\][/tex]
### Step 5: Clear the Fraction and Solve for [tex]\(x\)[/tex]
Multiply both sides by [tex]\(2\log x\)[/tex] to clear the fraction:
[tex]\[
\log 16 - 4\log 7 = 4\log x
\][/tex]
Now solve for [tex]\(\log x\)[/tex]:
[tex]\[
4\log x = \log 16 - 4\log 7
\][/tex]
Divide by 4:
[tex]\[
\log x = \frac{\log 16 - 4\log 7}{4}
\][/tex]
### Step 6: Determine [tex]\(x\)[/tex]
Calculate the numerical values (or leave in terms of logs if calculation not done):
- [tex]\(\log 16 = 4 \log 2\)[/tex]
- Using the given result, the value of [tex]\(x\)[/tex] that satisfies this equation is [tex]\(x = \frac{2}{7}\)[/tex].
Thus, the solution to the equation is [tex]\(x = \frac{2}{7}\)[/tex].
