Answer :
To solve the equation [tex]\(\log _5 3125 = \frac{t}{15}\)[/tex], we need to find the value of [tex]\(t\)[/tex].
1. Understand the logarithmic equation:
The equation [tex]\(\log _5 3125\)[/tex] means "the power to which 5 must be raised to get 3125."
2. Calculate the logarithm:
Determine [tex]\(\log _5 3125\)[/tex]. In this case, 3125 is a power of 5. Specifically, [tex]\(3125 = 5^5\)[/tex]. Therefore, [tex]\(\log _5 3125 = 5\)[/tex].
3. Set up the equation:
We know that:
[tex]\[
\log _5 3125 = \frac{t}{15}
\][/tex]
Substituting the value we found:
[tex]\[
5 = \frac{t}{15}
\][/tex]
4. Solve for [tex]\(t\)[/tex]:
To find [tex]\(t\)[/tex], we need to remove the fraction by multiplying both sides by 15:
[tex]\[
5 \times 15 = t
\][/tex]
[tex]\[
75 = t
\][/tex]
5. Conclusion:
The solution for [tex]\(t\)[/tex] is 75. Thus, the value of [tex]\(t\)[/tex] that satisfies the equation [tex]\(\log _5 3125 = \frac{t}{15}\)[/tex] is [tex]\(t = 75\)[/tex].
1. Understand the logarithmic equation:
The equation [tex]\(\log _5 3125\)[/tex] means "the power to which 5 must be raised to get 3125."
2. Calculate the logarithm:
Determine [tex]\(\log _5 3125\)[/tex]. In this case, 3125 is a power of 5. Specifically, [tex]\(3125 = 5^5\)[/tex]. Therefore, [tex]\(\log _5 3125 = 5\)[/tex].
3. Set up the equation:
We know that:
[tex]\[
\log _5 3125 = \frac{t}{15}
\][/tex]
Substituting the value we found:
[tex]\[
5 = \frac{t}{15}
\][/tex]
4. Solve for [tex]\(t\)[/tex]:
To find [tex]\(t\)[/tex], we need to remove the fraction by multiplying both sides by 15:
[tex]\[
5 \times 15 = t
\][/tex]
[tex]\[
75 = t
\][/tex]
5. Conclusion:
The solution for [tex]\(t\)[/tex] is 75. Thus, the value of [tex]\(t\)[/tex] that satisfies the equation [tex]\(\log _5 3125 = \frac{t}{15}\)[/tex] is [tex]\(t = 75\)[/tex].
