Step-by-step explanation:
Given the logarithmic expression:
[tex]3log_{5}2 + log_{5}x = log_{5}24[/tex]
Using the Logarithmic Product Rule:
[tex]log_{a}x + log_{a}y = log_{a}(xy)[/tex]
And the Logartihmic Power Rule:
[tex]log_{a}x^{y} = ylog_{a}x[/tex]
Rewriting your given logarithmic expression by applying the power rule:
[tex]log_{5}2^{3} + log_{5}x = log_{5}24[/tex]
I will use the following logarithmic property to cancel out the logarithmic expressions, which will show why x = 3.
[tex]_{b}log_{b}x = x[/tex]
Raise each logarithmic expression by the same value as the base, 5:
[tex](_{5}log_{5}2^{3} ) + (_{5}log_{5}x )= (_{5}log_{5}24)[/tex]
2³(x) = 24 (2³ is multiplied with x instead of addition because of the Product Rule).
Now that you're left with just 2³(x) = 24, it's easy to see why the answer is x = 3:
2³(x) = 24
8x = 24
Divide both sides by 8:
8x/8 = 24/8
x = 3
