solve exactly for a in the equation:

log a^5 + log a^(3/2) = log 625
4 2 8

the 4,2,8 are the bases

log4 a^5 + log2 a^(3/2) = log8 625

let's convert to the same base 2

log4 a^5 = log2 a^5 /log2 4
= (1/2)(log2 a^5) since log2 4 = 2

similary log8 625
= log2 625/log2 8

so the new equation is
(1/2)(log2 a^5) + log2 a^(3/2) = (1/3)log2625

using the rules of logs

log2 a^(5/2) + log2 a^(3/2) = log2 625^(1/3)

log2 a^4 = log2 625^(1/3)

a^4 = 625^(1/3)
a = 5^(1/3) or the cuberoot of 5

To solve for a in the equation:

log base 4 of a^5 + log base 2 of a^(3/2) = log base 8 of 625,

We can use the properties of logarithms to simplify the equation.

First, let's use the property that log base b of x + log base b of y = log base b of xy:

log base 4 of (a^5 * a^(3/2)) = log base 8 of 625

Next, we can simplify the expression inside the logarithm:
a^5 * a^(3/2) = a^(5 + 3/2) = a^(10/2 + 3/2) = a^(13/2)

Now, the equation becomes:
log base 4 of a^(13/2) = log base 8 of 625

Using another property of logarithms, we can rewrite the equation as:
(log base 4 of a) * (13/2) = (log base 8 of 625)

To simplify further, we can use the change of base formula to convert the equation to a common logarithm (base 10). The change of base formula states that log base b of x = log base c of x / log base c of b:

(13/2) * log base 10 of a = (log base 10 of 625) / (log base 10 of 4)

Now, we can solve for a by isolating it:

log base 10 of a = (log base 10 of 625) / (log base 10 of 4) * (2/13)

Finally, we can obtain the value of a by taking the antilog (exponentiation) of both sides:

a = 10^[(log base 10 of 625) / (log base 10 of 4) * (2/13)]

Using a calculator, you can evaluate this expression and obtain the value of a.