Answer :
Let's rewrite each equation as asked:
(a) Convert [tex]\(5^5 = 3125\)[/tex] to a logarithmic equation:
To express this exponential equation in logarithmic form, we use the relationship between exponentials and logarithms, which is:
If [tex]\(a^b = c\)[/tex], then [tex]\(\log_a(c) = b\)[/tex].
In this case:
- The base [tex]\(a\)[/tex] is 5.
- The exponent [tex]\(b\)[/tex] is 5.
- The result [tex]\(c\)[/tex] is 3125.
Applying this relationship, we rewrite [tex]\(5^5 = 3125\)[/tex] as:
[tex]\[
\log_5(3125) = 5
\][/tex]
(b) Convert [tex]\(\log 0.001 = -3\)[/tex] to an exponential equation:
This is a logarithmic equation using the common logarithm (base 10), as it doesn't specify a different base. To convert it to exponential form, we use the inverse relationship:
If [tex]\(\log_a(b) = c\)[/tex], then [tex]\(a^c = b\)[/tex].
Here:
- The base [tex]\(a\)[/tex] is 10 (since it's a common logarithm).
- The logarithm result [tex]\(c\)[/tex] is [tex]\(-3\)[/tex].
- The number [tex]\(b\)[/tex] is 0.001.
So, we rewrite the equation as:
[tex]\[
10^{-3} = 0.001
\][/tex]
These are your rewritings for each equation:
- Part (a): [tex]\(\log_5(3125) = 5\)[/tex]
- Part (b): [tex]\(10^{-3} = 0.001\)[/tex]
(a) Convert [tex]\(5^5 = 3125\)[/tex] to a logarithmic equation:
To express this exponential equation in logarithmic form, we use the relationship between exponentials and logarithms, which is:
If [tex]\(a^b = c\)[/tex], then [tex]\(\log_a(c) = b\)[/tex].
In this case:
- The base [tex]\(a\)[/tex] is 5.
- The exponent [tex]\(b\)[/tex] is 5.
- The result [tex]\(c\)[/tex] is 3125.
Applying this relationship, we rewrite [tex]\(5^5 = 3125\)[/tex] as:
[tex]\[
\log_5(3125) = 5
\][/tex]
(b) Convert [tex]\(\log 0.001 = -3\)[/tex] to an exponential equation:
This is a logarithmic equation using the common logarithm (base 10), as it doesn't specify a different base. To convert it to exponential form, we use the inverse relationship:
If [tex]\(\log_a(b) = c\)[/tex], then [tex]\(a^c = b\)[/tex].
Here:
- The base [tex]\(a\)[/tex] is 10 (since it's a common logarithm).
- The logarithm result [tex]\(c\)[/tex] is [tex]\(-3\)[/tex].
- The number [tex]\(b\)[/tex] is 0.001.
So, we rewrite the equation as:
[tex]\[
10^{-3} = 0.001
\][/tex]
These are your rewritings for each equation:
- Part (a): [tex]\(\log_5(3125) = 5\)[/tex]
- Part (b): [tex]\(10^{-3} = 0.001\)[/tex]
