Maths - Exponentials

Created: February 01, 2021

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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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