Sure, let's solve for [tex]\( x \)[/tex] from the equation
[tex]\[ \log(7x + 3) + 2 \log(5) = 2 + \log(x + 3) \][/tex]
Here's a detailed, step-by-step solution:
1. Simplify the logarithmic terms involving constants:
We know that [tex]\( a \log b = \log(b^a) \)[/tex]. Hence, [tex]\( 2 \log(5) = \log(5^2) = \log(25) \)[/tex].
Therefore, the equation becomes:
[tex]\[ \log(7x + 3) + \log(25) = 2 + \log(x + 3) \][/tex]
2. Combine the logarithmic terms on the left-hand side:
Using the property [tex]\( \log a + \log b = \log(ab) \)[/tex], we can combine [tex]\( \log(7x + 3) \)[/tex] and [tex]\( \log(25) \)[/tex]:
[tex]\[ \log((7x + 3) \cdot 25) = 2 + \log(x + 3) \][/tex]
This simplifies to:
[tex]\[ \log(175x + 75) = 2 + \log(x + 3) \][/tex]
3. Isolate the logarithmic expressions:
Next, we want to isolate the logarithmic terms on each side of the equation. Subtract [tex]\( \log(x + 3) \)[/tex] from both sides:
[tex]\[ \log(175x + 75) - \log(x + 3) = 2 \][/tex]
4. Combine the left-hand side using the properties of logarithms:
Using the property [tex]\( \log \frac{a}{b} = \log(a) - \log(b) \)[/tex]:
[tex]\[ \log \left( \frac{175x + 75}{x + 3} \right) = 2 \][/tex]
5. Exponentiate both sides to eliminate the logarithm:
Recall that if [tex]\( \log(a) = b \)[/tex], then [tex]\( a = 10^b \)[/tex]. Here, we use the natural logarithm base, [tex]\( e \)[/tex], which means [tex]\( e^{\log(a)} = a \)[/tex] and [tex]\( e^2 = e^2 \)[/tex].
[tex]\[ \frac{175x + 75}{x + 3} = e^2 \][/tex]
6. Solve the resulting algebraic equation:
Now we need to solve for [tex]\( x \)[/tex]. Set up the equation:
[tex]\[ 175x + 75 = e^2 (x + 3) \][/tex]
Expand and rearrange to get all terms involving [tex]\( x \)[/tex] on one side:
[tex]\[ 175x + 75 = e^2 x + 3e^2 \][/tex]
[tex]\[ 175x - e^2 x = 3e^2 - 75 \][/tex]
[tex]\[ x (175 - e^2) = 3e^2 - 75 \][/tex]
Finally, solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{3e^2 - 75}{175 - e^2} \][/tex]
Thus, the solution for [tex]\( x \)[/tex] in the given equation is:
[tex]\[ x = \frac{3(e^2 - 25)}{175 - e^2} \][/tex]
This result can also be simplified further but relies on the values calculated within the problem setting.
Therefore,
[tex]\[ x = \frac{3 \left(e^2 - 25 \right)}{175 - e^2} \][/tex]
is the final solution for [tex]\( x \)[/tex].
