Scientific Notation and Logarithms

A number is in scientific notation when written $a \times 10^n$ with $1 \le a < 10$ and $n$ an integer.

• Multiply: multiply the $a$ parts and add the exponents. Divide: divide and subtract.
• Renormalise at the end: $30 \times 10^{-3} = 3.0 \times 10^{-2}$ (the exponent goes up by one when $a$ is divided by 10).
• Square root: first make the exponent even. $\sqrt{4.9 \times 10^5} = \sqrt{49 \times 10^4} = 7 \times 10^2$.
• The order of magnitude is the power of ten that sets the size: $3.2 \times 10^6$ is of order $10^6$.

For a base $b > 0$, $b \ne 1$, and a number $x > 0$: $\log_b x$ is the exponent $y$ with $b^y = x$.

• $\log_b 1 = 0$, $\log_b b = 1$, $\log_b (b^k) = k$ and $b^{\log_b x} = x$.
• $\log x$ or $\log_{10} x$ is the common logarithm (base 10): $\log 1000 = 3$, $\log 0.01 = -2$.
• $\ln x$ is the natural logarithm, base $e \approx 2.718$: $\ln e = 1$, $\ln 1 = 0$.
• The logarithm of zero or of a negative number is not defined. A negative logarithm only means $0 < x < 1$.

For positive $x, y$ and any valid base:

• Product: $\log(xy) = \log x + \log y$.
• Quotient: $\log \frac{x}{y} = \log x - \log y$.
• Power: $\log x^k = k \log x$ (also for roots: $\log \sqrt{x} = \frac{1}{2}\log x$).
• Change of base: $\log_b x = \frac{\log x}{\log b}$, using any common base (10, $e$ or 2).
• There is no law for $\log(x + y)$.

• Common base: if both sides are powers of the same base, set the exponents equal.
• Take logs: $a^x = c$ gives $x = \frac{\log c}{\log a}$ for $c > 0$.
• $a^x$ is always positive, so $a^x = c$ with $c \le 0$ has no solution.
• Undoing a log relation: $\log y = k \log x + c$ means $y = 10^c\, x^k$; with $\ln$ it means $y = e^c x^k$.
• Harder equations (a substitution, or a logarithm of an expression in $x$) are in Equations and Inequalities.