Question

If log b(5) = 20 and log b (4)=13, then evaluate the following expression:

ln 125 / (ln b + ln 256) / ln b

Answer 1

The expression ln 125 / (ln b + ln 256) / ln b simplifies to 60 / 53 using logarithmic properties and the initial values given for log base b of 5 and 4.

Step-by-step explanation:

To evaluate the expression given the information that log base b of 5 is 20 and log base b of 4 is 13, we start by translating the logarithmic information into exponential form and then apply properties of logarithms to simplify the expression.

Given:

logb(5) = 20

logb(4) = 13

Therefore, we have:

b20 = 5

b13 = 4

The expression evaluates to:

ln(125) / (ln(b) + ln(256)) / ln(b)

Using the property that the logarithm of a product is the sum of the logarithms and the logarithm of a number raised to an exponent is the product of the exponent and the logarithm, we simplify the expression as follows:

ln(125) = ln(53) = 3 × ln(5)

Since b20 = 5, ln(5) = 20 × ln(b). Therefore, ln(125) = 3 × 20 × ln(b) = 60 × ln(b).

ln(256) = ln(44) = 4 × ln(4)

Since b13 = 4, ln(4) = 13 × ln(b). Therefore, ln(256) = 4 × 13 × ln(b) = 52 × ln(b).

So, the original expression becomes:

(60 × ln(b)) / ((ln(b) + 52 × ln(b)) / ln(b))

Simplifying further gives us:

(60 × ln(b)) / (1 + 52) = 60 × ln(b) / 53

Thus, the final result is:

60 / 53, since ln(b) cancels out.