Answer :
Sure, let's solve the given logarithmic equation step-by-step.
Given:
[tex]\[ \log_4(5x + 1) = 2 \][/tex]
1. Rewriting the logarithmic equation in exponential form:
The equation [tex]\(\log_b(y) = x\)[/tex] can be rewritten in exponential form as [tex]\(b^x = y\)[/tex].
So, [tex]\(\log_4(5x + 1) = 2\)[/tex] can be rewritten as:
[tex]\[ 4^2 = 5x + 1 \][/tex]
2. Simplify the equation:
Calculate [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Thus, the equation becomes:
[tex]\[ 16 = 5x + 1 \][/tex]
3. Isolate the term involving [tex]\(x\)[/tex]:
Subtract 1 from both sides:
[tex]\[ 16 - 1 = 5x \][/tex]
[tex]\[ 15 = 5x \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 5:
[tex]\[ x = \frac{15}{5} \][/tex]
5. Final answer:
[tex]\[ x = 3.0 \][/tex]
So, the solution for [tex]\(x\)[/tex] is [tex]\(3.0\)[/tex].
Given:
[tex]\[ \log_4(5x + 1) = 2 \][/tex]
1. Rewriting the logarithmic equation in exponential form:
The equation [tex]\(\log_b(y) = x\)[/tex] can be rewritten in exponential form as [tex]\(b^x = y\)[/tex].
So, [tex]\(\log_4(5x + 1) = 2\)[/tex] can be rewritten as:
[tex]\[ 4^2 = 5x + 1 \][/tex]
2. Simplify the equation:
Calculate [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Thus, the equation becomes:
[tex]\[ 16 = 5x + 1 \][/tex]
3. Isolate the term involving [tex]\(x\)[/tex]:
Subtract 1 from both sides:
[tex]\[ 16 - 1 = 5x \][/tex]
[tex]\[ 15 = 5x \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 5:
[tex]\[ x = \frac{15}{5} \][/tex]
5. Final answer:
[tex]\[ x = 3.0 \][/tex]
So, the solution for [tex]\(x\)[/tex] is [tex]\(3.0\)[/tex].
