Maths - Exponentials
See Also
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 $10^{\log _ {10} x}$?
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 $1$??
\[e\]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 $t$?
What do you get if you simlify the $\log _ 10$ of both sides of $p = ab^t$?
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?
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.