1. ## Logarithms

1.) log(x+1)^0.25 = 0.50

2.) 2^(x-1) = 3^(x-1)

3.) 600 = 1000(.04^(.3^x))

Your assistance would be much appreciated.

2. Hello, currypuff!

$\displaystyle 1)\;\;\log(x+1)^\frac{1}{4} \:= \:\frac{1}{2}$
I will assume that this log is base-ten.

We have: .$\displaystyle \frac{1}{4}\cdot\log(x+1) \:=\:\frac{1}{2} \quad\Rightarrow\quad \log(x+1) \:=\:2$

Then: .$\displaystyle x + 1 \:=\:10^2\quad\Rightarrow\quad \boxed{x \:=\:99}$

$\displaystyle 2)\;\;2^{x-1} \:= \:3^{x-1}$

Take logs: .$\displaystyle \ln\left(2^{x-1}\right) \:=\:\ln\left(3^{x-1}\right) \quad\Rightarrow\quad (x-2)\ln(2) \;=\;(x-1)\ln(3)$

. . $\displaystyle x\ln(2) - \ln(2) \:=\:x\ln(3) - \ln(3) \quad\Rightarrow\quad x\ln(2) - x\ln(3) \:=\:\ln(2) - \ln(3)$

Factor: .$\displaystyle x\left[\ln(2)-\ln(3)\right] \;=\;\ln(2)-\;\ln(3)$

Therefdore: .$\displaystyle x \;=\;\frac{\ln(2)-\ln(3)}{\ln(2)-\ln(3)}\quad\Rightarrow\quad\boxed{x \:=\:1}$

$\displaystyle 3)\;\;1000\left(0.04^{0.3^x}\right) \:=\:600$

Divide by 1000: .$\displaystyle 0.04^{0.3^x} \:=\:0.6$

Take logs: .$\displaystyle \ln\left(0.04^{0.3^x}\right) \:=\:\ln(0.6) \quad\Rightarrow\quad 0.3^x\cdot\ln(0.04) \:=\:\ln(0.6)$

. . Then we have: .$\displaystyle 0.3^x \;=\;\frac{\ln(0.6)}{\ln(0.04)}$

Take logs: .$\displaystyle \ln\left(0.3^x\right) \;=\;\ln\left[\frac{\ln(0.6)}{\ln(0.04)}\right] \quad\Rightarrow\quad x\ln(0.3) \;=\;\ln\left[\frac{\ln(0.6)}{\ln(0.04)}\right]$

$\displaystyle \text{Therefore: }\;x \;=\;\frac{\ln\left[\dfrac{\ln(0.06)}{\ln(0.04)}\right]}{\ln(0.3)}$

3. Originally Posted by currypuff

1.) log(x+1)^0.25 = 0.50
use the rules:

(1) $\displaystyle \log_a x^n = n \log_a x$

and

(2) $\displaystyle \log_a b = c \Longleftrightarrow a^c = b$

identify what are your a,b, and c and then just follow the formula

2.) 2^(x-1) = 3^(x-1)
log both sides.

you get $\displaystyle \ln 2^{x - 1} = \ln 3^{x - 1}$

then use law (1) from above and solve for x. remember, ln(2) and ln(3) are constants

3.) 600 = 1000(.04^(.3^x))
divide through by 1000

you get $\displaystyle 0.6 = 0.04^{0.3^x}$

log both sides, you get:

$\displaystyle \ln 0.6 = \ln 0.04^{0.3^x}$

apply law (1) above:

$\displaystyle \ln 0.6 = 0.3^x \ln 0.04$

divide by $\displaystyle \ln 0.04$, you get:

$\displaystyle \frac {\ln 0.06}{\ln 0.04} = 0.3^x$

log both sides again:

$\displaystyle \ln \left( \frac {\ln 0.6}{\ln 0.04} \right) = \ln 0.3^x$

now apply law (1) again and continue