[Math] Finding log base when given only a graph

logarithms

I have the function f(x)=log_bx is shown as a graph and the only two points are (5,1) and it's asking me to find the base of the logarithmic function.

Best Answer

Exponentiating both sides gives: $$ \log_b 5 = 1 \Rightarrow \\ b^{\log_b 5} = b^1 \Rightarrow \\ 5 = b $$

Related Solutions

Logarithms – How to Convert from Natural Log to Log Base 10

You do not have to memorize the change of base formula:

Say you want to evaluate $\log_{a}(x)$. Let's call it $y$. Then:

$$\begin{align} y = \log_{a}(x) &\iff a^y = x \\ & \iff \ln(a^y) = \ln(x) \\ & \iff y \ln(a) = \ln(x) \\ &\iff y = \frac{\ln(x)}{\ln(a)}. \end{align}$$

This shows that $\log_{a}(x) = \frac{\ln(x)}{\ln(a)}$, but hopefully this also emphasizes that you need not memorize the change of base formula in order to use it.

[Math] Writing an equation for a log function given the graph

You pick a functional form with some number of parameters. "A logarithmic function" is not a sufficient description. Seeing the vertical asymptote, I would guess th form you want is $y=a \log (3-bx)$ To get the point $(-1,0)$, you need $b=\frac 34$. Now find the $a$ that fits best. If that doesn't satisfy, try another form. If it sounds like a bit of art, it is. You can use root-finders for the parameters once you choose a form.

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Related Solutions

Logarithms – How to Convert from Natural Log to Log Base 10

[Math] Writing an equation for a log function given the graph

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