Help with This equation. Logs

[rawr]

I'm working with logs, and index's etc.
I have been given the question
solve for x:
5^(5-3x)=2^(x+2)

help, asap is appreciated.
thankyou.

 

[Denis]

Can't be solved directly, only numerically.

 

[galactus]

I think we can solve this algebraically.

$\displaystyle 5^{5-3x}=2^{x+2}$

$\displaystyle (5-3x)ln(5)=(x+2)ln(2)$

$\displaystyle \frac{5-3x}{x+2}=\frac{ln(2)}{ln(5)}$

$\displaystyle \frac{11}{x+2}-3=\frac{ln(2)}{ln(5)}$

Add 3, then reciprocal of each side:

$\displaystyle \frac{x+2}{11}=\frac{ln(5)}{ln(250)}$

Now, finish up?.

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Another way:

$\displaystyle \frac{5^{5}}{5^{3x}}=2^{2}\cdot 2^{x}$

$\displaystyle \frac{3125}{5^{3x}}=4\cdot 2^{x}$

$\displaystyle \frac{3125}{4}=(5^{3}\cdot 2)^{x}$

Now, finish?.

 

[Denis]

Nice "catch" Galactus. I musta been dreaming of Lady Gaga :?

 

[soroban]

Hello, rawr!

Another approach . . .

$\displaystyle \text{Solve for }x: \;\;5^{5-3x}\:=\:2^{x+2}$

$\displaystyle \text{Take logs: }\;\ln\left(5^{5-3x}\right) \:=\:\ln\left(2^{x+2}\right)$

. . . . . . .\(\displaystyle (5-3x)\!\cdot\!\ln5 \:=\x+2)\!\cdot\!\ln2\)

. . . . .$\displaystyle 5\!\cdot\!\ln5 - 3x\!\cdot\!\ln5 \:=\:x\!\cdot\!\ln2 + 2\!\cdot\!\ln2$

. . . . .$\displaystyle x\!\cdot\!\ln2 + 3x\!\cdot\!\ln5 \:=\:5\!\cdot\!\ln5 - 2\!\cdot\!\ln2$

. . . . .$\displaystyle x\!\cdot\!(\ln 2 + 3\ln 5) \:=\:5\!\cdot\!\ln5 - 2\!\cdot\!\ln 2$

. . . . . . . . . . . . . .$\displaystyle x \:=\:\frac{5\!\cdot\!\ln5 - 2\!\cdot\!\ln2}{\ln 2 + 3\!\cdot\!\ln5}$


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This can be simplified beyond all recognition . . .


. . $\displaystyle x \;=\;\frac{\ln(5^5) - \ln(2^2)}{\ln(5^3) + \ln2} \;=\; \frac{\ln3125 - \ln4}{\ln125 + \ln 2}$

. . . . $\displaystyle = \;\frac{\ln\left(\frac{3125}{4}\right)}{\ln(125\cdot2)} \;=\;\frac{\ln781.25}{\ln250}$

. . . . \(\displaystyle = \;1.206364638\hdots\)

 

[Denis]

Well, to make up for my faux pas (and before Subhotosh has me stand in the corner!) :

m^(ax + u) = n^(bx + v)

x = (u - kv) / (kb - a) where k = log(n) / log(m)

From the given problem: m=5, n=2, a=-3, b=1, u = 5, v = 2, x=?

 

[Deleted member 4993]

Nice "catch" Galactus. I musta been dreaming of Lady Gaga :?

Lady Gaga in your dreams - you are a brave man - her face scares the nightmare out of me...... :mrgreen: :mrgreen: