Manipulation of Indices

$8^m=(2^3)^{m} = 2^{3m}$

$4^m=(2^2)^{m} = 2^{2m}$

$2^{3m} = 27$

$\therefore 2^m = \sqrt[3]{27} = 3$

$2^{2m}= (2^m)^{2}$

$2^{2m} = 3^2 = 9$

$4^m = (8^{\frac{2}{3}})^m = (\color{#3D99F6}{8^m})^{\frac{2}{3}} = \color{#3D99F6}{27}^{\frac{2}{3}} = \boxed{9} \quad \quad \color{#3D99F6}{\text{It is given that }8^m = 27}$

To be honest, I actually used calculator for fast relieve.

4^(Ln 27/ Ln 8) = 9

$\large 8^m=27\\ \implies (2^m)^3= 3^3 \\\therefore (2^m)^2=3^2~~~~~\\Implies~~~~4^m=9.$

$8^m = 27 \\\\ \log_8 27 = m \\\\ m = \frac{1}{3} \cdot 3 \cdot \log_2 3 \\\\ \boxed{m = \log_2 3}$

Consideremos $4^m = k$ , segue que:

$\log_4 k = m \\\\ \frac{1}{2} \cdot \log_2 k = \log_2 3 \\\\ \log_2 k = 2 \cdot \log_2 3 \\\\ \log_2 k = \log_2 9 \\\\ k = 9 \\\\ \boxed{\boxed{4^m = 9}}$

Using excel: Set A1 equal to 8 Set B1 equal to 1 Set C1 equal to =A1^B1 Set A2 equal to 4 Set C2 equal to =A2^B1

Open the Solver add in, set the target cell to C1, which will have a value of 27. Allow it to change cell B1. Hit solve.

Done.

$8 = 2 \times 2 \times 2$

$8^{m} = ( 2 \times 2 \times 2 )^{m} = 27$

$2^{m} \times 2^{m} \times 2^{m} = 27$

if we took $x=2^{m}$ then

$3 \times x = 3 \times 9$

$x=9$ which means $2^{m} =9$

Log 8^m = Log 27

m Log 8 = Log 27

m = Log 27 / Log 8

m = 1.585

Press your calculator to get 4 raised to the power of 1.585

4^1.585 = 9

answer 9

It's a matter of reducing both sides to manipulate an answer here, without having to figure out the actual value of m. So, the problem offers us a base of 8 in one equation, and a base of 4 equalling the number that is the answer to this problem.

It is given that 8^M = 27, and again 4^M = ?

To reduce the base 8 to a similar base 4 in the other equation, knowledge and intuition would help guide in manipulating the 8 to be used as a 4; we have to take 8 to the power of a fraction that will output 4.

Knowing that 8 is 2 × 2 × 2, we can reduce this to 2^3. But, we also know that 2^3 still does not equal 4. Yes, we need to make the exponent 3 go under a numerator of 2 to manipulate further. Therefore, or in other words, the third root of 8 squared is 4.

8^(2/3) = 4 ==》 the base we want.

Awesome! So 4 can also be written, although not as friendly to the mind, or eyes, 8^(2/3) . Next, we must remember that if we reduce or manipulate in any way one side of an equation, the other side must have the same changing effect; The 8^(2/3 m) [ Remember we have that 'm' still in the exponent region] must also equal, symmetrically, 27^(2/3).

Alas, 8^(2/3 m) [OR as is starred above, 4^(m), and which is part of the second equation] = 27^(2/3) which reduces to a nice integer, 9.

ln(27)/ln(8)=m, 4^m=9

I didn't even solve for m. 8 is 4 to the 3/2th power. Finding 27 to the 2/3th power gives 9.

m log (8) = log (27) m = log (27)÷log (8) 4^(log (27)÷log (8)) is approximately 9

$8=4^ \frac{3}{2}$

$4^ \frac{3m}{2}=27$

$4^m=27^ \frac{2}{3}=9$

8^m = 27 so, (2^3)^m = 3^3 Then, it can be (2^m)^3 = 3^3 Finally, 2^m = 3 We can find 4^m, which is equivalent with (2^2)^m or (2^m)^2 It means that (2^m)^2 = (3)^2 = 9

4^log8(27)

By trial and error 8^1. 8549= ~27

So: 4^1.8549 =~9 Of course I used my calculator 🙂

8^{ m } = 27 ... Applying log... m log8 = log 27-------- (eq(1)) ....nd let 4^{ m } = x and apply log to this eq... =>m log(4) = logx------(eq(2)) solving (eq(1)) nd (eq(2)) (which we can do in seconds)... we get $\frac{log (27)}{log(x)}$ = $\frac{3}{2}$ so...

log(x) = log[(27)]^{ $\frac{2}{3}$ } = log(9) which gives us that x=9

Use the BAQ method for logs: Base to the answer = the question so log(8) of 27 = 1.54.... So m = 1.54... Then 4^1.54 = 9

m~1.5849624999 (by trying to guess the answer using the calculator) 4^(1.5849624999)~9

Not a good way of getting the answer, but it worked :)

8^m = 27
<=> m = ln(27)/ln(8) = 1.58...

=> 4^m = 4^1.58... = 9

4^log(8)27 = 9

\log_8 27 = 1.58 so 4^1.58 = ~ 9

8^m=27 8=(27)^(1/m) log(8)=(1/m)log(27) .9=(1/m)(1.43) (.9/.43)=1/m m=1/.63 m=1.6

4^m=9

4^m=p so m.ln(4)=ln(p),, 8=4^1.5,,27=9^1.5. 4^1.5m=9^1.5 ,,1.5m.ln4=1.5ln9. so m.ln4=ln9. Ln(p)=ln9 so,,,, p=4^m=9#####

First take log on both sides ... find m .... then put it in 4^m and get the answer...

Maybe this will be another solution: ${ 8 }^{ m }=27\\ { 8 }^{ m }={ 3 }^{ { 3 } }\\ m\ln { 8 } =3\ln { 3 } \\ m=3*\frac { ln{ 3 } }{ ln{ 8 } } \\ m=1.585\\ { 4 }^{ m }=9$

8^m = 27

ln (8^m) = ln (27)

m * ln (8) = ln (27)

m = ln (27) / ln (8)

4^(ln (27) / ln (8)) = 9

8^m = 27 => m = log2 (3) => 4^m = 2^log2 (3^2) =9

. 8^m = 27 .mlog(8)= log(27) . m=log(27)/log(8) . m=1.584962501 . 4^m=9

8^m = 27; 2^3m = 3^3 Therefore 2^m = 3, Then (2^m)^2 = 3^2: (2^2)^m = 9: ie 4^m = 9