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Logarithmic Functions

What are Logarithms? Laws of Logarithms Euler's Number, $e$ Natural Logarithms $ln$ Change of Base Logarithms to Solve Equations Logarithmic Functions

Extension Problems

If $21^x=2.1^y=0.01$, find the value of $\frac{1}{x}-\frac{1}{y}$.

If $2\log\left(a-b\right)=\log a+\log b$, find the ratio of $a:b$.
Write your answer in the form $\frac{a+\sqrt{b}}{c}:1$ where $a,b$ and $c$ are integers.

Solve for $x$ in the equation $\displaystyle 5^{x+1}+\frac{4}{5^x}=21$
Find the two values and separate them with a comma. If a value is not an integer, express using logs.

$x=$

Solve for $x$ in the equation $\displaystyle 3\times 9^x-2\times 4^x=5\times 6^x$
Give your answer in the form $\frac{\log{a}}{\log{b}-\log{c}}$ where $a,b$ and $c$ are integers.

$x=$

Using the fact that $\log 2 \approx 0.30103\ldots$, find how many digits $2^{1000}$ has.

Solve for $x$ in the equation $6\left(9^x+9^{-x}\right)-35\left(3^x+3^{-x}\right)+62=0$.
Find the four values and separate them with commas. If a value is not an integer, express using logs.

How many integer solutions satisfy the inequality $\displaystyle \frac{\log \left(x-1\right)}{\log 3}+\frac{\log \left(4x-7\right)}{\log 3}\le 3$?
(by Heesoo Jung)

Evaluate $\left(\log _43+\log _83\right)\left(\log _32+\log _92\right)$.

Solve the following: $\displaystyle \left(\log _4x\right)^2<\log _4x^2+8$

Evaluate $\displaystyle \log _{\sqrt{a}}\left(\sqrt{a\sqrt{a\sqrt{a\sqrt{a}}}}\right)$ where $a$ is a positive real number and $a\ne 1$.

Find all solutions for $x$ in the equation $\displaystyle x^{\log x}=100x$.
*Separate your answers with commas

Solve the following system of equations given $a \lt b$. Give answers using natural log ($\ln$) if needed.
$$ \left\{ \begin{array}{c} 2^{a+b}=56 \\ 2^a+2^b=15 \\ \end{array} \right. $$

$a=$

$b=$

Find the sum of all the prime factors of $N$ if $\displaystyle \log_{3}{\log_{5}{\left(\frac{N}{2}\right)}}=6$

Solve for $x$ in the equation $\displaystyle 8^x+2^x=130$
Hint: once you get a cubic equation, split the middle term and factorize (similar to splitting the middle term for quadratics)

$x=$

If $a>b>1$ and $\displaystyle \frac{1}{\log_ab}+\frac{1}{\log_ba}=\sqrt{293}$, find the value of $\displaystyle \frac{1}{\log_{ab}{b}}-\frac{1}{\log_{ab}{a}}$.

When $1 \lt a \lt b \lt a^2$, rank the following in order from least (1) to greatest (4):

$\log_a{b}$
$\log_b{a}$
$\log_a{ab}$
$\log_b{\frac{b}{a}}$