Exponents and Logarithms

Table of Contents

Exponentials

  • Exponentials are another type of function
  • Exponentials have the general form $P=e^x$, where $a$ is a constant

Euler’s Number $(e)$

  • In an exam, you’ll basically never get the easy version of exponentials

  • Instead you’ll get the Cool Kid version, which has this thing called Euler’s Number (also known as $e$)

    $e\approx 2.718281828459045235360…$ (it’s irrational, like $\pi$)

To understand $e$, imagine a bank that pays 100% interest over 1 year.

  • In 1 year, $1 becomes $2.

  • Now, let’s say you want even better interest. You go to bank 2, which pays 50% interest, twice per year.

    • At 6 months, you have $1.50.
    • At 1 year, you have $2.25.
    • Already you’re doing better!
  • You might remember the formula $A=(1+\frac{r}{n})^n$ from year 9 or 10. We can use this to find more profitable periods:

    • If your bank gives monthly payments, $A=(1+\frac{1}{12})^{12}=2.613…$
    • If your bank pays interest 10000 times a year, $A=(1+\frac{1}{10000})^{10000}=2.718…$
  • As you can see, even though the number of payments is increasing rapidly, the final amount is approaching the value of $e$.

  • This is because the formula for $e$ is $\displaystyle e=\lim_{n\rightarrow\infty} (1+\frac{1}{n})^n$, which is very similar to the compound interest formula.

  • Therefore, $e$ is defined as the rate of continous compounding interest.

Properties of $e$

  1. The gradient of the graph of $y=e^x$ is $e^x$ at every point on the graph

  2. The area under the graph of $e^x$ is $e^x$ at every point on the graph

  3. $e$ is the sum of the infinite series:

    • $\displaystyle e=\sum_{n=0}^{\infty}\frac{1}{n!}=\frac{1}{1}+\frac{1}{1}+\frac{1}{1\cdot2}+\frac{1}{1\cdot2\cdot3}+\frac{1}{1\cdot2\cdot3\cdot4}…$

Log Laws

  1. $a^b =c \leftrightarrow log_a c=b$

    Log is just another way of writing a power. It’s usually used to solve for $x$ when $x$ is in the power.

  2. $\log_e a=\ln a$

  3. $\log_a a=1$

  4. $log_a 1=a$

  5. $\color{lightblue}log_a b=\frac{ln(b)}{ln{a}}\color{lightgreen}\leftarrow\text{Change of Base Formula}$

  6. $log_a b+log_a c=log_a(b\cdot c)$

  7. $log_a b-log_a c=log_a(\frac{b}{c})$

  8. $log_a b^c =c\cdot\log_a b$

  9. $log_a a^x =x$

  10. 10.$a^{log_a b} =b$

  11. $\ln e^x =x$

  12. $e^{ln x}=x$


References

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Jackson Taylor
Jackson Taylor
Post Writer

I’m a Year 12 student at Sydney Tech.

Pranav Sharma
Pranav Sharma
Site Owner

Year 12 Student, site owner and developer.

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