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## Integrated math 3

### Course: Integrated math 3>Unit 6

Lesson 8: Graphs of logarithmic functions

# Graphical relationship between 2ˣ and log₂(x)

Sal graphs y=2ˣ and y=log₂(x) on the same coordinate plane, showing how they relate as graphs of inverse functions. Created by Sal Khan.

## Want to join the conversation?

• How were the exponential and logarithm functions invented?
• I've read that Leonhard Euler "discovered" log functions. I'm not sure if that means he recognized the pattern, or he just named them "logarithmic functions"
• How did you get 1
• Any number raised to power 0 = 1.
For example, 1^0 = 1
5^0 = 1
1000^0 =1.
This is one property of exponents. With y=2^x, if you check what is the value of y if x=0, then y=2^0=1. From this, you can also conclude that for any basis c (y=c^x), when x=0, => y=1.
• At , how did someone found out all this?
• As Sal says, exponential functions and logarithmic functions are inverses so they appear as reflections on the graph. Basically, the x values and y values are swapped.
• How do I find the Asymptote of a logarithmic function? Is there any formula for this?
• Logarithmic functions have vertical asymptotes, so follow the graph and see where it won't go past a line (ex. x equals 0 or the y-axis for the log in this video. Also, since log graphs are inverses of the exp graphs, an exp graph with a horizontal aymptote of the x axis would have a log asymptote of the y axis (I believe).
• How do you simplify log functions. or express them when (log...) is the exponent?
• You can think of log functions in the form of 'y=logbx', with b as the base, you switch the positions of the letters, making it 'x=b^y'. When switching from log to exponential forms, make it 'x=', make 'b' the base of the other side, and make 'y' the exponent.

I know it's been 2 years, but I hope this helps.
• Are logarithmic functions exponential?
• No. The logarithmic function is the inverse function of the exponential function. This is means that if a^x = b (exponential), then log base a (b) = x. (logarithmic). Therefore, exponential and logarithmic functions are not the same.
• How would you graph f=-log_2_(x)-4? I think it would be different from how exponential functions work.
• If it helps, you could rearrange the equation to be y+4=log_2(x), and then x=2^(y+4), then graph it as a function in terms of y.
If you don’t want to do the rearranging, plug in a few points (at x=1 and x=2 would be easy to calculate in your head) and then draw a smooth curve. The method is similar to an exponential function, but it does look different.
Let me know if you have further questions.
• Can logarithmic functions we written in the form y = a * (log_b(x-h)) + k just like parabolic functions can be written as y = a(x-h)^2 + k?