ESOL Solving Exponential and Logarithmic Equations Flashcard

Front

What is the definition of a logarithm?

Back

A logarithm is the exponent to which a base must be raised to produce a given number. For example, if b^y = x, then log_b(x) = y.

Front

What is the change of base formula for logarithms?

Back

The change of base formula states that log_b(a) = log_k(a) / log_k(b) for any positive k.

Front

What is the product rule for logarithms?

Back

The product rule states that log_b(mn) = log_b(m) + log_b(n).

Front

What is the quotient rule for logarithms?

Back

The quotient rule states that log_b(m/n) = log_b(m) - log_b(n).

Front

What is the power rule for logarithms?

Back

The power rule states that log_b(m^n) = n * log_b(m).

Front

How do you solve the equation log(x - 1) - log(x - 2) = log 5?

Back

Combine the logs using the quotient rule: log((x - 1)/(x - 2)) = log 5. Then set (x - 1)/(x - 2) = 5 and solve for x, yielding x = 9/4.

Front

What does it mean if a logarithmic equation has no solution?

Back

It means that the values of the variable do not satisfy the conditions of the logarithm, such as producing a negative argument.

Front

How do you condense the expression log(x) + log(y) + log(z^3)?

Back

Using the product and power rules, it condenses to log(xyz^3).

Front

What is the solution to the equation log_9(-5x) - log_9(6) = 1?

Back

Using the quotient rule, set -5x/6 = 9^1, leading to x = -10.800.

Front

How do you solve the equation 2(16^(x+6)) - 8 = 35?

Back

First, isolate the exponential term: 2(16^(x+6)) = 43. Then divide by 2 and solve for x, yielding x = -4.89.