# Maths - Exponentials

> Source: https://ollybritton.com/notes/a-level/maths/topics/exponentials/ · Updated: 2021-02-01 · Tags: maths, further-maths

## See Also
- [Maths - Differentiation](https://ollybritton.com/notes/a-level/maths/topics/differentiation/)

## Flashcards
##### $$y = a^x \\ y = a^{-x}$$ What is true about these two graphs??
They are reflections of each other in the $y$-axis.

##### $$y = a^x$$ What is the $y$-intercept of this graph??
$$
1
$$

##### $$\log_a b = c$$ If this is true, what is also true??
$$
a^c = b
$$

##### $$3^x = 9$$ What would you do to both sides to make $x$ the subject??
$$
\log_3
$$

##### $$\log_a a$$ What is this equal to??
$$
1
$$

##### $$\log_a 1$$ What is this equal to??
$$
0
$$

##### $$\log_a \frac{1}{a}$$ What is this equal to??
$$
-1
$$

##### $$\log_a m + \log_a n$$ How could you rewrite this??
$$
\log_a mn
$$

##### $$\log_a mn$$ How could you rewrite this??
$$
\log_a m + \log_b n
$$

##### $$\log_a m - \log_b n$$ How could you rewrite this??
$$
\log_a \left(\frac{m}{n}\right)
$$

##### $$\log_a \left(\frac{m}{n}\right)$$ How could you rewrite this??
$$
\log_a m - \log_b n
$$

##### $$\log_a x^n$$ How could you rewrite this??
$$
n \log_a x
$$

##### $$n \log_a x$$ How could you rewrite this??
$$
\log_a x^n
$$

##### $$\log_a \left(\frac{1}{y}\right)$$ How could you rewrite this??
$$
-\log_a y
$$

##### $$-\log_a y$$ How could you rewrite this??
$$
\log_a \left(\frac{1}{y}\right)
$$

##### $$2\log a$$ How could you rewrite this??
$$
\log a^2
$$

##### $$\frac{1}{2} \log a$$ How could you rewrite this??
$$
\log\sqrt{a}
$$

### 2021-02-02
##### Why does $\log_a x$ always cut the $x$-axis at $1$??
Because $a^0$ always equals $1$.

##### Why does the graph of $\log_a x$ get steeper the smaller value of $a$??
Because you have the raise $a$ to a higher power to get the same result.

##### What is $10^{\log_{10} x}$??
$$
x
$$

##### For what value of $a^x$ does the ratio between the gradient a point and the value of the point equal $1$??
$$
e
$$

##### $$\frac{dy}{dx} \div y : 2^x \to 0.7, 3^x \to 1.1$$ What value base do you need to raise to the power of $x$ for it to equal $1$??
$$
e
$$

### 2021-05-13
##### What's the general exponential model for a population $p$ with a initial population $a$, a "growth rate" $b$ and a time $t$??
$$
p = ab^t
$$

##### What do you get if you simlify the $\log_10$ of both sides of $p = ab^t$??
$$
\log_{10} p = t\log_{10} b + \log_{10} a
$$

##### What should you plot for a time $t$ and a poopulation size $p$ to see if the population grows exponentially??
$t$ against $\log_{10}(p)$.

##### What is the gradient of a $t$ against $\log_10(p)$ graph equal to??
$$
\log_{10}(b)
$$

### 2021-10-12
##### What is the first stage of solving $$3^{2x + 1} = 4^{3x}$$??
Taking any $\log$ of both sides.

##### How can you simplify this $$\ln(3^{2x+1}) = \ln(4^{3x})$$??
Using the power rule

$$
(2x+1)\ln(3) = (3x)\ln(3)
$$

### 2022-05-15
##### $$\ln(x)^2 - 2\ln(x) + 4$$ How can you prove this is never negative??
Complete the square.

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