# Exponential equation vs Logarithmic

1. Dec 12, 2005

### navneet

Hi I was wondering what is the relationship between y=a^x for exponential and y=loga (x) for log

I koe that we can divided this questiong further such as

a) equations --- which Im not sure

b) Graphs - I think we can say that the graph of the log equation is the reflection of exponential, and I stuck, also can we say something about which quads are they in??

please help me

2. Dec 12, 2005

### JasonRox

$log_a(x)$ is simply the inverse funtion of $a^x$.

To get the graph of $log_a(x)$, simply draw the graph of $a^x$ and do a reflection on the y=x line.

http://www.themathpage.com/aPreCalc/logarithmic-exponential-functions.htm

Look there for graphs of both functions.

All exponential functions intersect the y-axis at y=1, so they all have the point (0,1). Make sure you have that in the graph. We know this is true because...

Let... f(x) = a^x

Then... f(0) = a^0 = 1, for all a>0. Not sure for a=0.

Anyways, read along that page and let us know if you have more questions.

3. Dec 12, 2005

### HallsofIvy

Staff Emeritus
As JasonRox said, ax and loga x are inverse functions. That is, if y= ax, then x= loga y.

The graph of any bijective function, reflected in the y= x line, is the graph of the inverse function.

4. Dec 13, 2005

### VietDao29

a0 = 1 $\forall \ a \neq 0$...

5. Dec 13, 2005

### JasonRox

You must write according to the audience.

6. Dec 13, 2005

### motai

Adding on to what JasonRox said, in order to accurately graph the functions the bounds (domains/ranges and restrictions) must also be known. Since exponential equations and logarithmic equations are inverse functions, that means that the domain for the exponential is the range for the logarithmic, and vice versa.

In general, for the exponential function, the domain D is generally { D | -inf < x < +inf } with the range being { R | 0 < y < +inf }. The logarithmic function, is the exact opposite. For logarithmic equations, the domain D is { D | 0 < x < +inf } and the range is {R | -inf < y < +inf }. The bounds of the function will determine what quadrant the function is in (unless the function is shifted over [y=a^(x) - 1] or multiplied by a negative, such as y=-a^x).

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