Apply logarithm laws and inverse identities

What should happen first when condensing or expanding logarithms?

Try all positive inputs and compare both forms

Which conclusion fits every setting?

Logarithm laws

Logarithm laws change products, quotients and powers into simpler equivalent expressions. The base must be valid and every logarithm argument must be positive

1. 1Use the product law The logarithm of a product is the sum of the logarithms of its positive factors \(\displaystyle \log_{a}\left(xy\right)=\log_{a}\left(x\right)+\log_{a}\left(y\right)\)
2. 2Use the quotient law The logarithm of a quotient is the difference between the logarithms of its positive parts \(\displaystyle \log_{a}\left(\frac{x}{y}\right)=\log_{a}\left(x\right)-\log_{a}\left(y\right)\)
3. 3Use the power law An exponent on a positive logarithm argument becomes a coefficient outside the logarithm \(\displaystyle \log_{a}\left(x^{r}\right)=r\log_{a}\left(x\right)\)
4. 4Check inverse conditions Inverse cancellation is valid when the base is positive and not 1 and the logarithm argument is positive \(\displaystyle \log_{a}\left(a^{r}\right)=r,\quad a>0,\quad a\ne 1\)

Given

\(\displaystyle 8\times 4\)

First multiply the positive arguments before applying the product law

Given

\(\displaystyle \log_{2}\left(32\right)\)

Now evaluate the logarithm of the combined argument

Splitting a logarithm of a sum. One of these lines is right. Which one?

Produce a complete logarithmic expression

The power law moves a coefficient into an exponent. Addition combines products and subtraction combines quotients only when all original logarithms are defined. Preserve those original restrictions after simplification

Construct the full expression

Write L as a single logarithm and state its maximal real domain. Separately simplify R and retain its original excluded input. Explain why a shorter formula alone is not the same restricted function

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle R=\frac{x^2-25}{x-5}\)

Evaluate the simplified rational expression at x = −2.

Review the whole expression

Both logarithms require x>1. Condensation does not authorise negative original log arguments. For R, factor the numerator and cancel only for x≠5; the simplified line still has a hole there

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression

Produce the solution and filter its domain

First require x>1. Combine the logarithms, exponentiate, solve the quadratic and reject the negative candidate. Substitute the positive root in the original equation, not merely the transformed polynomial

Construct the full expression

Solve the equation by hand. Write every algebraic candidate, retain only the valid real solution and verify it in the original logarithms. Include the domain before using the log laws

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle \ln(x-2)+\ln(x+2)=\ln21\)

Enter the admissible solution

Review the whole expression

Both original logarithms require a positive argument. The negative candidate is outside the domain; substituting the positive candidate verifies the original equation

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression

State parameter conditions in a literal solution

For a>0, the real exponential can equal a and the logarithm gives the unique solution. If a≤0 there is no real solution. Parameter restrictions are part of the solution, not an optional note

Construct the full expression

Solve for x in terms of b. Classify b>0 and b≤0, give a substitution check, and explain how the negative coefficient of x affects the rearrangement

1. 1Your written solution Write the complete rule, derivation and domain on paper. Written practice is ungraded

Task

\(\displaystyle e^{3-2x}=b\)

For b=1, give the exact solution

Review the whole expression

For positive b, take logarithms and solve for x as shown above. For non-positive b, the exponential cannot equal b. Substituting the rule makes the exponent ln b.

1. 1Review criteria Check every term and sign, the original domain and any excluded inputs, then independently substitute or differentiate to verify. Explain discrepancies and correct your own written expression