1) write the expression as a single logarithm log3 40 - log3 10 show work. 2) solve the exponential equation 1/16 = 64^[4x-3]

1) The expression log3 (40) - log3 (10) can be simplified. When substracting logs with the same base values, in this case 3, you can simplify the expression by dividing the species inside each log.

log3 (40) - log3 (10)

log3 (40/10)

log3 (4) = simplified expression

2) To solve the exponential equation 1/16 = 64^ (4x-3) we can also simplify things first. 16 = 4², and 64 = 4³.

1/4² = 4³ ⁽⁴ˣ⁻³⁾

4⁻² = 4³ ⁽⁴ˣ⁻³⁾

At this stage since the bases are equal, the exponents must also be equal.

-2 = 3 (4x-3)

-2/3 = 4x - 3

4x = 7/3

x = 7/12

1) According to log rules, loga b - loga c=loga (b/c). Therefore, log 3 40-log3 10=log 3 (40/10) = log3 4.

2) 1/16=64^ (4x-3), take the logarithm of both sides. log (1/16) = (4x-3) log (64). log (1/16) = - log (16), so - log (16) = (4x-3) log (64). log (64) = log (4^4) = 4log (4), log (16) = log (4^2) = 2log (4), so log (64) = 2log (16). Therefore, (4x-3) = - 1/2, x=5/8.