I need some help in how to solve some of these logrithmic equeations?

Question

Im trying to figure out how to solve these problems

1) 2log(6) 4-1/3 log(6) 8=log(6)x

2) 2.3^(x^2)=66.6

3) e^2x-5e^x+6=0

4) ln 19.8 = ln e^0.083x

This on is tough
5) 4^x + 6(4^x)=5

The powere rule, quotient and product rules were what we were suposed to use.

Any help is much appriciated
Im just trying to figure out how to crack these step by step

Answer 1

1) 2log(6) 4-1/3 log(6) 8=log(6)x

I assume log(6) is "log to the base 6."

log(6) 4^2 + log(6) 8^(-1/3) = log(6) x
log(6) ( ( 4^2 )( 8^(-1/3 ) ) = log(6) x
(4^2) 8^(-1/3) = x

2) 2.3^(x^2)=66.6

2.3 ^ x = sqrt ( 66.6 )
x ln ( 2.3 ) = ln ( sqrt ( 66.6 ) )
x = ln ( sqrt ( 66.6 ) ) / ln ( 2.3 )
x = ln ( sqrt ( 66.6 ) - 2.3 )

3) e^2x-5e^x+6=0

(e^2)(e^x) - (5)(e^x) + 6 = 0
(e^2 - 5)(e^x) + 6 = 0
e^x = -6 / (e^2 - 5)
x = ln ( -6 / ( e^2 - 5 ) )

4) ln 19.8 = ln e^0.083x

ln ( 19.8 ) = .083 x
ln ( 19.8 ) / .083 = x

This on is tough

5) 4^x + 6(4^x)=5

( 1 + 6 ) 4^x = 5
4^x = 5 / 7
x ln ( 4 ) = ln ( 5 / 7 )
x = ln ( 5 / 7 ) / ln ( 4 )
x = ln ( 5 / 7 - 4 )

Answer 2

1) assuming base 6,
2log(6) 4-1/3 log(6) 8=log(6)x
log (6)4^2 - log (6) 8^1/3 = log (6)x
log (6) 16 - log (6) 2 = log (6)x
log (6) (16/2) = log (6)x
8 = x

2) take log of both sides so log 2.3 ^(x^2) = log 66.6
x^2 * log 2.3 = log 66.6
x^2 = log 66.6/ log 2.3
x = sqrt of log 66.6/log 2.3


3) factor :e^2x-5e^x+6=0
(e^x)^2 -5 * (e^x) + 6 = 0
(e^x - 6) (e^x +1) = 0
e^x = 6 e^x = -1
take log of both sides of each equation
log e^x = log 6 log e^x = log -1
x * log e = log 6 x * log e = log -1
x = log 6/ log e x = log -1/ log e


4) ln 19.8 = ln e^ 0.083x
ln 19.8 = 0.083x * ln e
0.083 x = ln 19.8/ ln e
x = (ln 19.8/ln e) / 0.083


5) 4^x + 6(4^x) = 5
7 (4^x) = 5
(4^x) = 5/7
log (4^x) = log 5/7
x* log 4 = log 5/7
x = log 5/7 / log 4


hope this helps!