Maths - Exponentials
Pearson Edexcel Mathematics 2022
See Also
Flashcards
Exponential graphs
\[y = a^x \\ y = a^{-x}\]
What is true about these two graphs?
They are reflections of each other in the $y$-axis.
Logarithms as inverses of exponentials
The laws of logarithms
\[\log _ a m - \log _ a n\]
How could you rewrite this?
\[\log _ a \left(\frac{m}{n}\right)\]
How could you rewrite this?
\[\log _ a \left(\frac{1}{y}\right)\]
How could you rewrite this?
\[-\log _ a y\]
How could you rewrite this?
Properties of the logarithm graph
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.
The number e
For what value of $a^x$ does the ratio between the gradient a point and the value of the point equal $1$?
\[\frac{dy}{dx} \div y : 2^x \to 0.7, 3^x \to 1.1\]
What value base do you need to raise to the power of $x$ for it to equal $1$?
Exponential population models
What’s the general exponential model for a population $p$ with a initial population $a$, a “growth rate” $b$ and a time $t$?
What do you get if you simplify the $\log _ {10}$ of both sides of $p = ab^t$?
What should you plot for a time $t$ and a population 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?
Solving equations with logarithms
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
\[(2x+1)\ln(3) = (3x)\ln(3)\]\[\ln(x)^2 - 2\ln(x) + 4\]
How can you prove this is never negative?
Complete the square.