Question

Given `1/4 log a^3  +  5 log sqrt(b) = 1`, find the value of a³b¹⁰.

Answer 1

`1/4 log a^3  +  5 log sqrt(b) = 1`

We need to find the value of a³b¹⁰.

Step 1: Simplify each logarithmic term:

`1/4 log a^3 = log a^(3/4)`

`5 log sqrt(b) = log b^(5/2)`

So the equation becomes:

`log a^(3/4)  +  log b^(5/2) = 1`

Step 2: Combine the logarithms:

Using the logarithmic property log a + log b = log (ab), we get:

`log(a^(3/4) xx b^(5/2)) = 1`

Step 3: Convert to exponential form:

Since log 10 = 1, we know:

`a^(3/4) xx b^(5/2) = 10`

Step 4: Solve for a³b¹⁰:

We want to find a³b¹⁰. 

First, express the terms to match the powers we need:

`a^(3/4) xx b^(5/2) = 10`

Raise both sides of the equation to the power of `4/3` to make the exponents on a and b match a³ and b¹⁰:

`(a^(3/4) xx b^(5/2))^(4/3) = 10^(4/3)`

Simplify:

`a^3 xx b^10 = 10^(4/3)`