# Logarithmic functions problems with answers

Recent questions in Logarithmic Functions
Bierlehre59 2022-08-14

### Do the logarithmic rules work when taking logs of functions as opposed to numbers?i.e. suppose f is a function and n is a real number, is $\mathrm{log}\left(f\left(x{\right)}^{n}\right)=n·\mathrm{log}\left(f\left(x\right)\right)$?

sondestiny120g 2022-08-11

### We know that:${e}^{x}=\sum _{n=0}^{\mathrm{\infty }}\frac{{x}^{n}}{n!}$$\mathrm{log}\left(x\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n+1}\left(x-1{\right)}^{n}}{n}$Where second series converges when |x−1|<1.It is possible to prove that:${e}^{\mathrm{log}\left(x\right)}=x$for $|x-1|<1$ using only series representation?

logosdepmpe 2022-08-11

### use the laws of logarithms to expand the expression${\mathrm{log}}_{3}\left(\frac{x\left({x}^{2}+5\right)}{\sqrt{{x}^{2}-5}}\right)$

Garrett Sheppard 2022-08-10

### I'm confused trying to figure out the domain of the logarithmic function below:$f\left(x\right)=\mathrm{ln}\left({e}^{x}+3\right)$Because the argument of $f,\phantom{\rule{thinmathspace}{0ex}}{e}^{x}+3,$, is a nonnegative number, ${e}^{x}+3>0$ and ${e}^{x}>-3.$ Taking natural logarithms on both sides, we get $x>\mathrm{ln}\left(-3\right)$. However, the domain of a logarithmic function must be nonnegative real numbers, so $\mathrm{ln}\left(-3\right)$ doesn't make sense. How then to determine the scope of the original function?

granuliet1u 2022-08-10

### I have the following function and I am trying to isolate x:$\mathrm{ln}\left(x\right)+\mathrm{ln}\left(x-1\right)=1$I take ex of both sides:$x+x-1=e$$2x–1=e$$2x=e+1$$x=\frac{e+1}{2}$

Ledexadvanips 2022-08-06

### The sum of series $\frac{\left(\mathrm{log}3{\right)}^{1}}{1!}+\frac{\left(\mathrm{log}3{\right)}^{3}}{3!}+\frac{\left(\mathrm{log}3{\right)}^{5}}{5!}+\cdots$ is what? Is there a general algorithm to find the summation of logarithms?

Braylon Lester 2022-07-22

### I just wonder can we call ${2}^{x}+x$ an exponential function or not?With the same way of thinking is ${\mathrm{log}}_{2}\left(x\right)+x$ a logarithmic?

scherezade29pc 2022-07-21

### Help me with thisIs this the same as

Patricia Bean 2022-07-20

### Let $aϵ{\mathbb{R}}_{+}$ and $b=\mathrm{exp}\left(-a\right).$What is the range of $\left\{-{b}^{2}\mathrm{log}\left(b\right)\right\}$? Does it range $\left(-\mathrm{\infty },+\mathrm{\infty }\right)$?How can I show $\frac{-b\mathrm{log}\left(b\right)}{1-b}\le 1$ ?

Emmanuel Pace 2022-07-20

### $y=\left(3{x}^{2}+2{\right)}^{lnx}$The answer to this is:$\frac{dy}{dx}=\left(3{x}^{2}+2{\right)}^{lnx}\left(\frac{1}{x}ln\left(3{x}^{2}+2\right)+\frac{6xlnx}{3{x}^{2}+2}\right)$What I'm coming up with is:$\frac{dy}{dx}=\left(3{x}^{2}+2{\right)}^{lnx}\left(\frac{1}{x}ln\left(3{x}^{2}+2\right)+\frac{6x}{3{x}^{2}+2}\right)$What I'm not understanding is where the $\frac{6xlnx}{3{x}^{2}+2}$ comes from, if anyone could explain this I'd really appreciate it.

Rishi Hale 2022-07-18

### A point on a graph is (1/8,−3) of the logarithmic function $f\left(x\right)=\mathrm{log}{b}^{x}$, and the point (4,k) is on the graph of the inverse, $y={f}^{-1}\left(x\right)$. Determine the value k.

Nash Frank 2022-07-17

### a)Express the given quantity as a single logarithm.$\frac{1}{3}\mathrm{ln}\left(x+2{\right)}^{3}+\frac{1}{2}\left[\mathrm{ln}\left(x\right)-\mathrm{ln}\left({x}^{2}+3x+2{\right)}^{2}\right]$b) Solve each equation for x.1)${e}^{5-4x}=4$x=?2) $\mathrm{ln}\left(3x-13\right)=8$x=?

Patricia Bean 2022-07-17

### I know that fraction denominator needs to be > 0 , so if denominator is quadractic equation i know how to solve , But since denominator is > 0 , I dont know how to solve , can anyone help me ?$f\left(x\right)=\sqrt{\mathrm{log}\left(\frac{3x-{x}^{2}}{2}\right)}$

pliwraih 2022-07-16