To solve for [tex]\( x \)[/tex] in the equation [tex]\( \log_2(5x - 7) = 1 \)[/tex], follow these steps:
1. Understand the given equation:
We start with the logarithmic equation:
[tex]\[
\log_2(5x - 7) = 1
\][/tex]
2. Convert the logarithmic equation to its exponential form:
The logarithmic equation [tex]\( \log_b(y) = x \)[/tex] is equivalent to the exponential equation [tex]\( y = b^x \)[/tex].
Thus,
[tex]\[
5x - 7 = 2^1
\][/tex]
3. Calculate the value of [tex]\( 2^1 \)[/tex]:
[tex]\[
2^1 = 2
\][/tex]
Therefore,
[tex]\[
5x - 7 = 2
\][/tex]
4. Isolate [tex]\( x \)[/tex]:
First, add 7 to both sides of the equation to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
5x - 7 + 7 = 2 + 7
\][/tex]
Simplifying this, we get:
[tex]\[
5x = 9
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Finally, divide both sides by 5:
[tex]\[
x = \frac{9}{5}
\][/tex]
Simplifying the fraction, we obtain:
[tex]\[
x = 1.8
\][/tex]
So the solution to the equation [tex]\( \log_2(5x - 7) = 1 \)[/tex] is:
[tex]\[
x = 1.8
\][/tex]
