Laws of Exponents
When multiplying powers with the same base, add the exponents. When dividing, subtract them. A power raised to a power multiplies the exponents. Any non-zero number to the power zero is one. A negative exponent means the reciprocal. A fractional exponent one over n means the n-th root. If a to the x equals a to the y, then x equals y.
Logarithms
The logarithm of M to base a equals x means a to the x equals M. The rules are: the log of a product is the sum of the logs; the log of a quotient is the difference; the log of M to the n equals n times the log of M; the log of a to base a is one and the log of one is zero. The base change rule says log of a to base b equals log a divided by log b in any common base. Also log of a to base b times log of b to base a equals one.
Key values with base 10: log 2 is 0.3010, log 3 is 0.4771, and log 5 is 1 minus log 2, which is 0.6990.
Unit Digit Cycles
Numbers ending in 0, 1, 5 or 6 keep the same last digit for every power. The powers of 2 end in 2, 4, 8, 6 repeating. The powers of 3 end in 3, 9, 7, 1. The powers of 7 end in 7, 9, 3, 1. Divide the exponent by 4 and use the remainder. If the remainder is zero, take the last digit in the cycle.
Shortcuts
In an exponent equation, bring both sides to the same base and equate the exponents. Learn the powers of 2 and 3 by heart. After solving a log equation, check that every logarithm has a positive argument.
Worked Examples
Example 1: Solve 2 to the x equals 32.
Solution: Since 32 is 2 to the fifth, x equals 5.
Example 2: Evaluate log to base 2 of 64.
Solution: Sixty-four is 2 to the sixth, so the answer is 6.
Example 3: Solve log x plus log of x minus 3 equals 1.
Solution: x times x minus 3 equals 10, so x squared minus 3x minus 10 is zero, giving x equals 5 or minus 2. Minus 2 is rejected, so x is 5.
Quick Check
Question 1: What is a to the power zero?
Answer: One, when a is not zero.
Question 2: What is the log of a product?
Answer: The sum of the logs.
Question 3: What is the unit digit cycle of powers of 3?
Answer: 3, 9, 7, 1.