Question 126382
Solve:(A) (1/27)sqrt3^x=81(^x-1)/9 
OK, it should look this then:
{{{(1/27)sqrt(3^x)}}} = {{{(81^(x-1))/9}}}
Multiply both sides by 27, gets rid of both denominators:
{{{sqrt(3^x) = 3(81^(x-1))}}}
{{{3^(x/2) = 3(81^(x-1))}}};  exponent equivalent of Square Root is 1/2
Use nat logs:
{{{ln(3^(x/2)) = ln(3) + ln(81^(x-1))}}};
Use log equiv of exponents
{{{(x/2)*ln(3) = ln(3) + (x-1)*ln(81)}}};
:
{{{(x/2)*(1.0986) = 1.0986 + (x-1)*(4.3944)}}}
Divided 2 into 1.0986
.5493x = 1.0986 + 4.3944x - 4.3944
:
.5493x = 4.3944x - 3.2958
:
.5493x - 4.3944x = -3.2958
:
-3.8451x = -3.2958
x = {{{(-3.2958)/(-3.8451)}}}
x = +.857
:
:
Check solution using calc
enter: {{{(1/27)sqrt(3^.857)}}} = .0593
and
{{{(81^(.857-1))/9}}} = .0593 also,  confirms our solution of x=.857
:
:
(B) (cubedsqrt4/2)^6x=2sqrt2
On the 2nd one is this what you mean?
{{{(cubeRt(4)/2)^(6x)}}} = {{{2sqrt(2)}}}
Use a calc to find cube root of 4 divided by 2: enter 4^(1/3)/2 = .7937
Use a calc to find the square root of 2 times 2: enter 2^(1/2)*2 = 2.8284
now we have:
{{{.7937^(6x) = 2.8284}}}
Find the nat log of both sides, use the log equiv of exponents
6x * ln(.7937) = ln(2.8284)
:
6x * -.23105 = 1.0397
:
-1.3863x = 1.0397; multiplied by 6
x = {{{1.0397/(-1.3863)}}}
x = -.74998 ~ -.75 is what it is:
:
Check this using a calc
Find the value of 6x: -.75*6 = -4.5
enter: {{{(4^(1/3)/2)^(-4.5)}}} = 2.8284
enter: {{{2sqrt(2)}}} = 2.8284 also confirms our solution x = -.75
:
Did this make sense to you, Any questions?