# Logarithms

• Feb 24th 2008, 11:09 AM
currypuff
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.
• Feb 24th 2008, 11:32 AM
Soroban
Hello, currypuff!

Quote:

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

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

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

Quote:

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

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

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

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

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

Quote:

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

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

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

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

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

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

• Feb 24th 2008, 11:38 AM
Jhevon
Quote:

Originally Posted by currypuff

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

use the rules:

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

and

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

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

Quote:

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

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

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

Quote:

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

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

log both sides, you get:

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

apply law (1) above:

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

divide by $\ln 0.04$, you get:

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

log both sides again:

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

now apply law (1) again and continue