What is the difference between log and ln?

log with no written base usually means base 10; ln means base e, where e is about 2.718. They obey identical laws, but they give different numerical answers and ln is the one that appears in continuous growth models and in calculus derivatives.

Why do I get a wrong answer even though my algebra was right on a log equation?

Almost always an extraneous solution. Condensing two logs into one widens the domain, so a root that solves the condensed equation can make the original log's argument zero or negative. Every solution has to be substituted back into the original equation.

Do we cover exponential growth and decay word problems, not just the algebra?

Yes. Half-life, doubling time, population and interest problems are worked through in sessions, including choosing between the (1+r)^t and e^(rt) forms and solving for the time variable using logs.

Is this the same material as the exponentials in calculus?

It is the algebra that comes before it. Derivatives and integrals of e^x and ln x sit in the Intro to Calculus topic; here the focus is manipulation, equation solving, graphs and modelling, which those calculus rules assume you can already do.

Logarithms & Exponentials

High School (11-12) · Mathematics

Logarithms are the inverse of exponentials, and most of the difficulty at this level comes from that single idea being used in several different disguises: as an algebraic rule set, as a tool for solving equations where the unknown sits in the exponent, and as a modelling language for growth, decay and compound interest. This topic covers both directions — reading log statements as exponent statements, and rewriting exponential relationships as logs — plus the natural base e and ln, which show up in almost every calculus and science course that follows. Sessions are spoken one-on-one with an AI tutor, so you talk through each rewrite step rather than watching a worked example.

Start a session on Logarithms & Exponentials

What this covers

Converting fluently between b^x = y and log_b(y) = x, including base 10, base e and awkward bases like 2 and 1/3
Applying the product, quotient and power laws to condense or expand expressions, and using change of base to evaluate log_5(80) on a calculator
Solving exponential equations by taking logs of both sides, including 3^(2x-1) = 40 and equations needing the same-base method first
Solving logarithmic equations and checking for extraneous roots against the domain of each log term
Graphing y = a·b^(x-h) + k and y = log_b(x-h), identifying horizontal or vertical asymptotes, domain, range and intercepts
Setting up and solving growth, decay, half-life and compound interest models, including finding the time variable and converting between A = P(1+r/n)^(nt) and A = Pe^(rt)

Where learners get stuck

Treating log(a + b) as log a + log b, or log(a)/log(b) as log(a - b): The product and quotient laws create a pattern of 'logs turn operations into simpler operations', and learners over-generalise it to addition inside the argument. Reinforcing that the laws only apply to products and quotients *inside* one log — and testing with numbers like log(10+90) — makes the boundary concrete.
Accepting every algebraic solution to a log equation without checking domains: Combining logs before solving changes the domain of the equation, so a perfectly valid quadratic root can produce log of a negative number. Learners rarely see the check modelled as part of the method, so it feels like an optional extra rather than a required step.
Confusing where the base goes, so log_2(8) is read as 'log times 2 times 8' or answered as 4: Logarithm notation hides its operation — there is no visible exponent — and subscripts are unfamiliar. Saying every log statement out loud as 'what power of 2 gives 8?' rebuilds the missing structure.
Believing ln and log are interchangeable, or that e is just another arbitrary constant: Calculators label them separately but many textbook problems work with either, so the distinction seems cosmetic until continuous-growth models and change of base produce different numerical answers.

What a session looks like

A typical session starts with two or three quick conversion or evaluation prompts to check the exponential-log link is secure, then moves to a set of equations that build in difficulty — same-base first, then taking logs, then equations needing the laws to condense before solving. You say each step aloud and the tutor asks why a particular law applies before letting you move on. Word problems (half-life, doubling time, interest) are worked by naming the variables first, then choosing the model, then solving for the exponent. Graph work is described verbally: asymptote, direction, intercepts, transformations.

Helpful to know first

Confident work with exponent rules, including negative and fractional exponents
Solving linear and quadratic equations, including factoring and the quadratic formula
Function notation, inverse functions, domain and range
Reading and sketching graphs with transformations (shifts, stretches, reflections)
Calculator use for log, ln and powers

Questions

What is the difference between log and ln?: log with no written base usually means base 10; ln means base e, where e is about 2.718. They obey identical laws, but they give different numerical answers and ln is the one that appears in continuous growth models and in calculus derivatives.
Why do I get a wrong answer even though my algebra was right on a log equation?: Almost always an extraneous solution. Condensing two logs into one widens the domain, so a root that solves the condensed equation can make the original log's argument zero or negative. Every solution has to be substituted back into the original equation.
Do we cover exponential growth and decay word problems, not just the algebra?: Yes. Half-life, doubling time, population and interest problems are worked through in sessions, including choosing between the (1+r)^t and e^(rt) forms and solving for the time variable using logs.
Is this the same material as the exponentials in calculus?: It is the algebra that comes before it. Derivatives and integrals of e^x and ln x sit in the Intro to Calculus topic; here the focus is manipulation, equation solving, graphs and modelling, which those calculus rules assume you can already do.