Exponential functions with any base, and exponential equations
Fill in a table of 2^x, see that a base above 1 gives a rising curve and a base between 0 and 1 a falling one, and solve an exponential equation by writing both sides with one base.
- Germany: grade 10
- Iran: grade 11
- Palestine: grade 12
- Ukraine: grade 11
- Russia: grade 11
Guide
The exponential function is y = a^x with a positive and a not 1. It works for negative x too: 2−1 = 12 and 20 = 1.
If a > 1 the graph rises from left to right and stays above the x-axis. If 0 < a < 1 it falls; y = (12)x is the mirror image of y = 2x in the y-axis. Both pass through (0, 1).
To solve an exponential equation write both sides as powers of one number: 4x = 8 becomes 22x = 23, so 2x = 3.
Common slips: 2^-1 = −2; reading a falling curve as growth; solving 4^x = 8 as 8 ÷ 4.