log5 +log20

log√8 ÷ log8

To solve these logarithmic expressions, we can use logarithmic properties and basic rules of logarithms.

1. log5 + log20:
To add logarithms with the same base, we can use the product rule of logarithms. The product rule states that logₐ(x) + logₐ(y) = logₐ(xy).

In this case, we have log5 + log20. Applying the product rule, we can rewrite this as log5(20). This means that log5 + log20 is equivalent to log₅(20).

So, log5 + log20 = log₅(20).

2. log√8 ÷ log8:
To simplify this expression, we can use the quotient rule of logarithms. The quotient rule states that logₐ(x) - logₐ(y) = logₐ(x/y).

In this case, we have log√8 ÷ log8. Rewriting the square root (√) in terms of exponents, we get log(8^(1/2)) ÷ log8.

Using the quotient rule, we can simplify this expression to log(8^(1/2) / 8).

Simplifying further, we have log(√8 / 8) which is log(2 / √8).

To rationalize the denominator, we can multiply both the numerator and denominator by √8 (square root of 8).

This gives us log(2 * √8 / √8 * √8), which simplifies to log(2√8 / 8).

Further simplifying the expression, we have log(2√8) - log(8).

Using properties of logarithms, we can rewrite this as 1/2 * log8(2) - 1.

So, log√8 ÷ log8 = 1/2 * log8(2) - 1.