To convert the given exponential equation [tex]\(10^w = c\)[/tex] into its logarithmic form, we need to use the properties of logarithms.
The general form of an exponential equation is [tex]\(a^b = c\)[/tex], where [tex]\(a\)[/tex] is the base, [tex]\(b\)[/tex] is the exponent, and [tex]\(c\)[/tex] is the result. To convert this into logarithmic form, we write it as:
[tex]\[
b = \log_a(c)
\][/tex]
Here is the step-by-step transformation:
1. Identify the base of the exponential equation, which is 10 in this case.
2. Identify the exponent, which is [tex]\(w\)[/tex].
3. Identify the result, which is [tex]\(c\)[/tex].
Given the exponential equation:
[tex]\[
10^w = c
\][/tex]
According to the properties of logarithms, we'll convert the equation into its equivalent logarithmic form. The logarithmic form:
[tex]\[
w = \log_{10}(c)
\][/tex]
Since the base of the logarithm is 10, we typically write it simply as:
[tex]\[
w = \log(c)
\][/tex]
So, the equation [tex]\(10^w = c\)[/tex] in logarithmic form is:
[tex]\[
w = \log(c)
\][/tex]
And this is the detailed, step-by-step solution to converting the given exponential equation into its logarithmic form.
