To solve for [tex]\( x \)[/tex] in the given equation:
[tex]\[
\frac{7.4 |x| 10^{-3}}{5.5 |x| 10^{-3}} = \frac{k (0.10)^2 (0.20)^x}{k (0.10)^2 (0.10)^x}
\][/tex]
let's simplify both sides of the equation step-by-step:
### Step 1: Simplify the Left Side
Firstly, observe that [tex]\( |x| \)[/tex] and [tex]\( 10^{-3} \)[/tex] are present in both the numerator and the denominator, so they can be canceled out. This leaves us with:
[tex]\[
\frac{7.4}{5.5}
\][/tex]
### Step 2: Simplify the Right Side
On the right side of the equation, notice that [tex]\( k \)[/tex] and [tex]\( (0.10)^2 \)[/tex] appear in both the numerator and the denominator, so they can be canceled out as well. This simplifies the right side to:
[tex]\[
\frac{(0.20)^x}{(0.10)^x}
\][/tex]
We can further simplify this as follows:
[tex]\[
\frac{(0.20)^x}{(0.10)^x} = \left(\frac{0.20}{0.10}\right)^x = 2^x
\][/tex]
### Step 3: Combining the Simplified Parts
With the left and right sides simplified, we are now left with the equation:
[tex]\[
\frac{7.4}{5.5} = 2^x
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Next, we determine the numerical value of [tex]\(\frac{7.4}{5.5}\)[/tex]:
[tex]\[
\frac{7.4}{5.5} \approx 1.3455
\][/tex]
Now, we have the equation:
[tex]\[
1.3455 = 2^x
\][/tex]
To solve this equation for [tex]\( x \)[/tex], we take the logarithm of both sides. Using the base-2 logarithm (logarithm base 2), we get:
[tex]\[
\log_2(1.3455) = x
\][/tex]
Alternatively, using the natural logarithm (base [tex]\( e \)[/tex], denoted as [tex]\( \ln \)[/tex]) and the change of base formula [tex]\(\log_b(a) = \frac{\ln(a)}{\ln(b)}\)[/tex]:
[tex]\[
x = \frac{\ln(1.3455)}{\ln(2)}
\][/tex]
### Step 5: Calculate [tex]\( x \)[/tex]
Using a calculator to find the natural logarithms:
[tex]\[
\ln(1.3455) \approx 0.296883702
\][/tex]
[tex]\[
\ln(2) \approx 0.693147181
\][/tex]
Then:
[tex]\[
x \approx \frac{0.296883702}{0.693147181} \approx 0.428
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x \approx 0.428
\][/tex]
