Laws of logarithms

The logarithm laws are exponent laws written in reverse. If $u=\log_a x$ and $v=\log_a y$, then $x=a^u$ and $y=a^v$. Therefore $xy=a^{u+v}$, so $\log_a(xy)=u+v=\log_a x+\log_a y$.

Similarly, $\frac{x}{y}=a^{u-v}$ gives the quotient law. If $x=a^u$, then $x^r=a^{ru}$, giving $\log_a(x^r)=r\log_a x$ whenever the expressions are defined.

These derivations explain why multiplication becomes addition, division becomes subtraction, and an exponent becomes a coefficient. They also explain why there is no corresponding rule for $\log_a(x+y)$: exponent laws contain no rule that turns a sum of powers into a single power.

Logarithm laws convert multiplication into addition, division into subtraction, and powers into coefficients. They follow from the corresponding exponent laws.

The product law applies to a product inside one logarithm. The quotient law applies to a quotient. The power law moves an exponent on the entire argument to the front.

There is no logarithm law for a sum or difference. Expressions such as $\log_a(x+y)$ and $\log_a(x-y)$ generally cannot be split.

Every separate logarithm created by expansion must have a positive argument. When variables appear, use the domain conditions supplied in the question or state appropriate restrictions.

Expanding rewrites one logarithm as several simpler terms. Condensing reverses the process and combines several logarithms into one. Coefficients must become exponents before product or quotient laws are used.

The change-of-base formula lets a GDC evaluate logarithms in any valid base using $\log$ or $\ln$. Both numerator and denominator must use the same new base.

Calculator answers are approximations unless the logarithm simplifies exactly. Retain full precision internally and round only at the final step.

For a real logarithm $\log_a(g(x))$, require $a>0$, $a\ne1$, and $g(x)>0$. If an equation contains several logarithms, every argument must be positive simultaneously.

Restrictions should be written before logarithms are combined or removed. Algebra may produce candidates outside the original domain, and those candidates must be rejected by substitution into the original equation.

The familiar power law $\log_a(x^r)=r\log_a x$ is stated with $x>0$. A useful HL refinement is $\log_a(x^2)=2\log_a|x|$ for $x\ne0$; writing $2\log_a x$ would incorrectly exclude negative $x$.

A base between $0$ and $1$ is valid. In this case $\log_a x$ is decreasing, so inequalities reverse when converting between logarithmic and exponential forms.

For exponential equations, first try to express both sides with the same base. If that is impossible, isolate the exponential expression and take logarithms.

For logarithmic equations, state domain restrictions, combine logarithms when useful, convert to exponential form, solve the resulting equation, and check every candidate in the original equation.

Taking logarithms is valid only after the isolated exponential side is positive. This is automatically true for $a^{f(x)}$ when $a>0$, but coefficients and other terms still need careful algebra.

A GDC can solve equations numerically using a solver or graph intersections. An IB response should still show the equation entered, the chosen interval or graph, and suitable rounding.

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.

Logarithms can transform some nonlinear models into straight-line form. For $y=Ab^x$, taking natural logarithms gives $\ln y=\ln A+x\ln b$. A plot of $\ln y$ against $x$ is linear with gradient $\ln b$ and vertical intercept $\ln A$.

For a power model $y=Ax^n$, taking logarithms gives $\ln y=\ln A+n\ln x$. A plot of $\ln y$ against $\ln x$ is linear with gradient $n$ and intercept $\ln A$. Common logarithms work equally well if the same base is used throughout.

After fitting a transformed line, convert the intercept back to the original parameter: if the intercept is $c$, then $A=e^c$ for natural logs or $A=10^c$ for common logs. Interpret the gradient and intercept in the original context.

A strong model check examines the transformed scatter plot and residuals, then considers whether constant percentage growth or a power relationship is reasonable. A high correlation does not prove the model will remain reliable beyond the observed data.

Use $\log$ for base $10$ and $\ln$ for base $e$. For any other valid base, enter $\log_a x=\frac{\ln x}{\ln a}$ or use the calculator's dedicated logarithm-base template.

For a numerical equation, graph the two sides and find every intersection in the stated domain, or use a numerical solver with sensible starting values. Scan the graph first because one initial estimate may locate only one of several roots.

When working with transformed data, create lists for the original variables and calculated lists for their logarithms. Run the linear regression on the transformed lists, record the regression equation, and back-transform its intercept before writing the original model.

Keep guard digits and round once at the end. In an IB response, write the equation or regression entered, report relevant coordinates or parameters, include units, and interpret the result in context.

These questions mix exact algebra, numerical technology, modelling, and interpretation. Show domain restrictions and exact working before using decimal approximations.

For application questions, identify the model and time unit, keep full calculator precision, apply any integer or domain constraint, and finish with a sentence that answers the question in context.

Use this checklist to choose a method quickly. First identify the logarithm base and all argument restrictions; then decide whether the task needs conversion, a law, an equation method, a model, or technology.

Exact algebra and contextual interpretation are equally important. A correct calculator value is not a complete IB AI HL answer unless its setup, accuracy, units, constraints, and meaning are clear.