This page covers the following topics:
1. Basics of exponentials
2. The exponential function
3. Laws of indices
4. Solving equations with exponents
An exponential function such as aˣ = b can be solved by rewriting it as logab = x, where a is the base of the logarithm. The logarithm can now be inputted on a calculator to solve for x. If the base of the logarithm is not written, it should be assumed to be 10.
The natural exponential function is y = eˣ, where e is Euler's number. The inverse of the natural exponent is the natural logarithm, lnx. This can be used to solve equations with the two functions.
All laws of indices apply when using the natural exponential function: when multiplying, the indices are added, and when dividing, the indices are subtracted. When Euler's number is raised to the power of 0, it equates to 1.
Equations involving eˣ can be solved using the laws of indices and the knowledge that the natural logarithm is its inverse.
Find the unknown value a of y-coordinate of the point given on the diagram. Give the answer to 3 significant figures.
y(7) = e⁷ = 1100 (3 s. f.)
Solve log(4x + 2) = 2.
log(4x + 2) = 2
4x + 2 = 10² = 100
4x = 98
x = 24.5
x = 24.5
Solve 5eˣ = 2eˣ + 9.
Rearranging gives 3eˣ = 9, so eˣ = 3.
Taking the natural logarithm of both sides gives x = ln3.
x = ln3
Simplify e⁵ × e⁶/e².
e⁵ × e⁶/e² = e¹¹/e² = e⁹
The x-coordinate value of point A is −2. Find the y-coordinate value of A to 2 decimal points.
y(−2) = e⁻² = 0.14 (2 d. p.)
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