1. ## Logarithmic Functions

Could someone please solve these questions for me? Thanks

2. ## Re: Logarithmic Functions

Originally Posted by strangepath
Could someone please solve these questions for me? Thanks
First, we are not a homework service. Please post what work you've been able to do on these.

-Dan

3. ## Re: Logarithmic Functions

Hello, strangepath!

That's quite a list, but I see that they require a variety of tecnniques.
I'll solve a few of them.

$\displaystyle \text{(1) Solve for }n\!:\;\;11^{n+1} \:=\:13$

Take logs: .$\displaystyle \ln(11^{n+1}) \:=\:\ln(13) \quad\Rightarrow\quad(n+1)\ln(11) \:=\:\ln(13)$

. . . . . . . . .$\displaystyle n+1 \:=\:\frac{\ln(13)}{\ln(11)} \quad\Rightarrow\quad n \:=\:\frac{\ln(13)}{\ln(11)} - 1$

$\displaystyle \text{(4) Solve for }x\!:\;\;5^{x+1} - 5^x \:=\:8$

This one is deceptively simple.

Factor: .$\displaystyle 5^x(5-1) \:=\:8 \quad\Rightarrow\quad 5^x\cdot4 \:=\:8 \quad\Rightarrow\quad 5^x \:=\:2$

Take logs: .$\displaystyle \ln(5^x) \:=\:\ln(2) \quad\Rightarrow\quad x\ln(5) \:=\:\ln(2) \quad\Rightarrow\quad x \:=\:\frac{\ln(2)}{\ln(5)}$

$\displaystyle \text{(7) Solve for }x\!:\;\;10^{4\log x} \:=\:16$

We are expected to know that: .$\displaystyle 10^{4\log x} \:=\:10^{\log x^4} \:=\:x^4$

The equation becomes: .$\displaystyle x^4 \:=\:16 \quad\Rightarrow\quad x \:=\:2$

. . . We must discard x = -2.

$\displaystyle \text{(8) Solve for }w\!:\;\;3e^w - 35e^{-w} \:=\:-16$

Multiply by $\displaystyle e^w\!:\;\;3e^{2w} - 35 \:=\:-16e^w \quad\Rightarrow\quad 3e^{2w} + 16e^w - 35 \:=\:0$

Factor: .$\displaystyle (e^w + 7)(3e^w - 5) \:=\:0$

And we have two equations to solve:

. . $\displaystyle \begin{Bmatrix}e^w + 7 \:=\:0 & \Rightarrow & e^w \:=\:\text{-}7 &\Rightarrow & \text{no real roots} \\ \\ 3e^w - 5 \:=\:0 & \Rightarrow& e^w \:=\:\frac{5}{3} & \Rightarrow & w \:=\:\ln(\frac{5}{3}) \end{Bmatrix}$

4. ## Re: Logarithmic Functions

Originally Posted by topsquark
First, we are not a homework service. Please post what work you've been able to do on these.