Exponents and introductory 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.

The base, exponent and result do not change when we switch forms: $2^3=8\Longleftrightarrow\log_2(8)=3$.

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.

An exponential model has the form $Q(t)=Q_0b^t$, where $Q_0$ is the initial value and $b$ is the multiplier per time period. Growth has $b>1$; decay has $0<b<1$. A rate of $r\%$ gives $b=1+\frac{r}{100}$ for growth and $b=1-\frac{r}{100}$ for decay.

Always match the time variable to the rate period. If a population grows by $8\%$ each hour, then $t$ must count hours in $Q=Q_0(1.08)^t$. If time is given in minutes, convert it to hours before substituting unless the model defines a minute-by-minute multiplier.

Logarithms answer inverse questions in a model. Substitution finds the quantity at a known time; solving $Q_0b^t=T$ finds the time required to reach a target: $t=\frac{\ln(\frac{T}{Q_0})}{\ln b}$, provided all quantities are positive and $b\ne1$.

A mathematical answer still needs contextual interpretation. A time of $4.41$ years may mean approximately $4.41$ years for a continuous model, but if a value is checked only once per whole year, test years $4$ and $5$ and report the first year that satisfies the condition. Exponential models also assume a constant multiplier; this may become unrealistic over long intervals.

Logarithmic scales compress quantities that vary across many powers of $10$. A change of $1$ unit on a base-10 logarithmic scale represents multiplication by $10$ in the original quantity. This is why logarithms are useful for acidity, sound intensity and earthquake measurements.

The pH model is $\mathrm{pH}=-\log_{10}[\mathrm H^+]$, where $[\mathrm H^+]$ is the hydrogen-ion concentration in moles per litre. Because of the minus sign, a lower pH corresponds to a larger hydrogen-ion concentration. A difference of one pH unit represents a factor of $10$ in concentration.

The decibel model $L=10\log_{10}(\frac{I}{I_0})$ compares intensity $I$ with a reference intensity $I_0$. The model matters: logarithms compare ratios, so the ratio must be positive and unitless. In any application, state what the model output represents rather than reporting an unexplained number.

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.

For a model such as $Q=Q_0b^t$, use a table to inspect whole-number times and a graph or solver for a continuous-time estimate. Set a window that includes both the model and the target line. An intersection gives the boundary value; the wording of the problem determines which side of that boundary is valid.

A reliable technology check uses two methods when possible. For example, solve $450(1.065)^t=900$ with $t=\frac{\ln2}{\ln1.065}$, then confirm by graphing $y_1=450(1.065)^x$ and $y_2=900$. Agreement supports the calculation but does not replace units or interpretation.

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.

For exponents, identify the base and the operation before choosing a law. Products add same-base exponents, quotients subtract them, and a power of a power multiplies them. A zero exponent requires a non-zero base; a negative exponent means reciprocal.

For logarithms, translate the question into “what exponent?” Use $a^x=b\Longleftrightarrow\log_a b=x$, with $a>0$, $a\ne1$, and $b>0$. Match bases for exact equations; otherwise isolate the power and use logarithms or technology.

For applications, write the model, identify units and time periods, calculate with guard digits, then interpret. A decimal time, a whole-period threshold and a long-term forecast are different kinds of answers.