Integer exponents and introduction to logarithms

The exponent laws are consequences of counting equal factors. When powers with the same non-zero base are multiplied, add their exponents. When they are divided, subtract the denominator exponent from the numerator exponent. These rules work for positive, zero, and negative integer exponents whenever the expressions are defined.

A power raised to another power multiplies the exponents. A product or quotient raised to a power applies that power to every factor in the brackets. The same-base condition is essential for the product and quotient laws: $2^3\times5^3$ cannot be simplified by adding the exponents, although it can be written as $(2\times5)^3=10^3$.

Exponent laws change products, quotients, and powers—not sums. In general, $(a+b)^n\ne a^n+b^n$. For example, $(2+3)^2=25$, while $2^2+3^2=13$.

A strong simplification method is to work factor by factor. Expand only when it helps, collect numerical coefficients, combine each repeated base using exponent laws, and then rewrite negative exponents as reciprocals if the question asks for positive exponents.

An algebraic expression with a negative exponent is not automatically wrong. However, IB questions often request an answer with positive exponents. Move a factor across the fraction bar to change the sign of its exponent: $\frac{x^{-4}}{y^{-2}}=\frac{y^2}{x^4}$.

Restrictions should be carried with algebra involving division or negative exponents. If the original expression contains $\frac{1}{x}$, then $x\ne0$, even if later cancellation removes the visible denominator.

A logarithm answers one precise question: what exponent must the base be raised to in order to obtain this number? The statement $\log_a b=c$ means exactly the same thing as $a^c=b$. Logarithms and exponentiation are inverse operations, just as subtraction reverses addition.

For real logarithms, the base must satisfy $a>0$ and $a\ne1$, and the argument must satisfy $b>0$. A positive base never produces zero or a negative result when raised to a real power, so $\log_a 0$ and $\log_a(-5)$ are undefined in the real number system.

Two bases are especially common. The common logarithm $\log x$ usually means $\log_{10}x$. The natural logarithm $\ln x$ means $\log_e x$, where $e\approx2.71828$. Both are calculator functions and both obey the same logarithm laws.

Logarithm laws turn multiplication into addition, division into subtraction, and powers into coefficients. They follow directly from the exponent laws. For example, if $x=a^m$ and $y=a^n$, then $xy=a^{m+n}$, so $\log_a(xy)=m+n=\log_a x+\log_a y$.

Every logarithm argument involved in a rule must be positive, and the base must be positive and not equal to 1. There is no general identity $\log_a(x+y)=\log_a x+\log_a y$. The expressions can coincide for special values, but logarithms do not generally distribute over addition or subtraction.

The power rule is useful in both directions. It can expand $\log_a(x^k)$ to $k\log_a x$, or condense $k\log_a x$ to $\log_a(x^k)$, provided the domain conditions are satisfied.

First look for a common base. If both sides can be written as powers of the same valid base, equate the exponents. This gives an exact solution and is usually the fastest non-calculator method.

If a common base is not convenient, isolate the exponential expression and take a logarithm of both sides. Either $\ln$ or $\log$ works because the change-of-base ratio cancels consistently. Keep full calculator precision until the final answer.

In contextual problems, the algebraic value may need interpretation. If time is counted in complete years or cycles and a quantity must first exceed a target, test the neighbouring integers rather than simply rounding the logarithmic answer.

For a single logarithm, convert directly to exponential form. For several logarithms with the same base, first condense them using the logarithm laws, then convert to exponential form. Solve the resulting algebraic equation and check every candidate in the original logarithms.

Domain checks are part of the solution. A candidate that makes any logarithm argument zero or negative must be rejected, even if it satisfies the transformed polynomial equation.

When logarithms appear on both sides with the same one-to-one base, equal logarithms of valid positive arguments imply equal arguments: $\log_a f(x)=\log_a g(x)\Rightarrow f(x)=g(x)$, subject to both domain conditions.

Use the $x^y$, power, or caret key for general powers. Enter negative exponents with brackets when needed: type $5^{(-3)}$, not $-5^3$. Many calculators also distinguish the negative-number key from the subtraction key.

Use the $\log$ key for base $10$ and the $\ln$ key for base $e$. To evaluate another base, use change of base: $\log_a b=\frac{\log b}{\log a}=\frac{\ln b}{\ln a}$. Never mix bases within the same ratio.

For a numerical exponential or logarithmic equation, a GDC graph or numerical solver can confirm the algebraic result. Define the two sides as functions and find their intersection, or enter the equation into the solver with a sensible initial estimate. In an IB solution, record the equation and relevant calculator result rather than writing only a decimal answer.

Retain guard digits. Store or reuse the full calculator value and round once at the end to the requested number of decimal places or significant figures.

The following mixed questions bring together exponent laws, logarithmic notation, equation solving, domains, and calculator accuracy. Attempt each question without reading the steps first, then compare your working line by line.

In an IB response, exact values should remain exact unless a decimal is requested. For numerical answers, show the equation, keep full precision during working, and attach the requested accuracy to the final line.