Answer :
Sure! Let's rewrite the given logarithmic equation [tex]\(\log_4(8) = \frac{3}{2}\)[/tex] in exponential form. Here is the step-by-step solution:
1. Understand the logarithmic form:
The logarithmic equation [tex]\(\log_b(a) = c\)[/tex] means that [tex]\(b^c = a\)[/tex], where [tex]\(b\)[/tex] is the base, [tex]\(a\)[/tex] is the result, and [tex]\(c\)[/tex] is the exponent.
2. Identify the components:
In the given equation [tex]\(\log_4(8) = \frac{3}{2}\)[/tex], we can identify:
- The base [tex]\(b\)[/tex] is 4.
- The result [tex]\(a\)[/tex] is 8.
- The exponent [tex]\(c\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
3. Rewrite in exponential form:
Using the relationship [tex]\(b^c = a\)[/tex], we substitute [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(a\)[/tex] with 4, [tex]\(\frac{3}{2}\)[/tex], and 8, respectively:
[tex]\[ 4^{\frac{3}{2}} = 8. \][/tex]
Thus, the logarithmic equation [tex]\(\log_4(8) = \frac{3}{2}\)[/tex] can be rewritten in exponential form as [tex]\(4^{\frac{3}{2}} = 8\)[/tex].
1. Understand the logarithmic form:
The logarithmic equation [tex]\(\log_b(a) = c\)[/tex] means that [tex]\(b^c = a\)[/tex], where [tex]\(b\)[/tex] is the base, [tex]\(a\)[/tex] is the result, and [tex]\(c\)[/tex] is the exponent.
2. Identify the components:
In the given equation [tex]\(\log_4(8) = \frac{3}{2}\)[/tex], we can identify:
- The base [tex]\(b\)[/tex] is 4.
- The result [tex]\(a\)[/tex] is 8.
- The exponent [tex]\(c\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
3. Rewrite in exponential form:
Using the relationship [tex]\(b^c = a\)[/tex], we substitute [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(a\)[/tex] with 4, [tex]\(\frac{3}{2}\)[/tex], and 8, respectively:
[tex]\[ 4^{\frac{3}{2}} = 8. \][/tex]
Thus, the logarithmic equation [tex]\(\log_4(8) = \frac{3}{2}\)[/tex] can be rewritten in exponential form as [tex]\(4^{\frac{3}{2}} = 8\)[/tex].
