Certainly! Let's transform the given equation [tex]\(\log (\alpha \beta) = 5x^2 + 8y^2\)[/tex] into an exponential form.
1. Understand the Logarithmic Equation:
The given equation is:
[tex]\[
\log (\alpha \beta) = 5x^2 + 8y^2
\][/tex]
This can be interpreted as a logarithm with base 10 if no base is specified.
2. Rewrite Using the Definition of Logarithms:
Recall that the equation [tex]\(\log_b(a) = c\)[/tex] is equivalent to [tex]\(a = b^c\)[/tex]. Here the base [tex]\(b\)[/tex] is assumed to be 10.
3. Apply the Exponential Definition:
According to the definition, [tex]\(\log (\alpha \beta) = 5x^2 + 8y^2\)[/tex] implies that:
[tex]\[
\alpha \beta = 10^{5x^2 + 8y^2}
\][/tex]
Thus, the equation [tex]\(\log (\alpha \beta) = 5x^2 + 8y^2\)[/tex] can be rewritten in exponential form as:
[tex]\[
\alpha \beta = 10^{5x^2 + 8y^2}
\][/tex]
So, the solution in the required form is:
[tex]\[
\alpha \beta = 10^{5 \cdot x^2 + 8 \cdot y^2}
\][/tex]
Thus, we have:
[tex]\[
\alpha \beta = 10^{5x^2 + 8y^2}
\][/tex]
