Exponential and Logarithmic Equations

Solve the following exponential equations by writing both sides as powers of the same base:

1. a) $2^{x} = 8$

1. b) $3^{x} = \frac{1}{9}$

1. c) $5^{x} = 25$

1. d) $10^{x} = 0.001$

Solve the following exponential equations with the unknown in an exponent expression:

1. a) $3^{x + 1} = 27$

1. b) $2^{3x} = 16$

1. c) $5^{2 - x} = 125$

By applying the definition of a logarithm, solve:

1. a) $\log_{2} x = 5$

1. b) $\log_{3} x = -2$

1. c) $\log x = 4$

1. d) $\ln x = 0$

Solve by applying the appropriate inverse function:

1. a) $\ln x = 4$

1. b) $e^{x} = 7$

1. c) $10^{x} = 45$

1. d) $\log x = 2.5$

Solve the equation $4^{x} - 5 \times 2^{x} + 4 = 0$ using the substitution $t = 2^{x}$. Check the solutions obtained.

Solve the equation $2^{x + 1} + 2^{x} + 2^{x - 1} = 28$ by factorising out the smallest power. Check the solution.

Solve the exponential equation $9^{x} - 4 \times 3^{x} + 3 = 0$ by following these steps:

1. a) Express $9^{x}$ as a power with base $3$ and use the substitution $t = 3^{x}$.

1. b) Solve the resulting quadratic equation.

1. c) Reverse the substitution, check each solution in the original equation, and discard any that are not valid.

Solve the equations:

1. a) $\log_{2} x + \log_{2} 4 = 5$

1. b) $\log_{5} x - \log_{5} 2 = 2$

Solve the logarithmic equation $\log_{2}(x - 1) + \log_{2}(x + 1) = 3$ and check the validity of each candidate solution.

Solve the equation $\log x + \log(x + 3) = 1$, where $\log$ denotes the base-10 logarithm. Why must one of the solutions to the associated quadratic equation be discarded?

A student solves the equation $\log_{2}(x - 1) + \log_{2}(x - 3) = 3$ as follows:

'$\log_{2}[(x - 1)(x - 3)] = 3 \Rightarrow (x - 1)(x - 3) = 8 \Rightarrow x^{2} - 4x - 5 = 0 \Rightarrow x = 5$ and $x = -1$. Therefore, the equation has two solutions: $x = 5$ and $x = -1$.'

1. a) Explain where the error in their conclusion lies, even though the calculations are correct.

1. b) Write out the full and correct solution, stating which check is mandatory for every logarithmic equation.

Solve the following equations, in which the unknown appears in unusual places:

1. a) $\log_{x} 8 = \frac{3}{2}$ (the unknown is the base of the logarithm; remember which values a base can take).

1. b) $\log_{3}(x + 2) - \log_{3}(x - 2) = 1$ (apply the rule for the logarithm of a quotient and check the solution).

Solve the equations by taking base-10 logarithms on both sides. Express the exact solution as a quotient of logarithms (there is no need to give the decimal value):

1. a) $2^{x} = 15$

1. b) $7^{x} = 100$

Solve the equation $3^{x - 1} = 2^{x}$ by taking natural logarithms on both sides. Express the exact solution in terms of logarithms and give its approximate value to two decimal places.

The figure shows the curve $y = 2^{x}$ and the straight line $y = 5$.

1. a) Explain why the solution to the equation $2^{x} = 5$ is the x-coordinate of the intersection point labelled '?'.

1. b) Bound the solution between two consecutive integers using the graph or by trial and error.

1. c) Express the exact solution using a logarithm and calculate its approximate value to three decimal places.

A family deposits $\$100\,000$ into a savings account at $4 \%$ annual compound interest, so that the capital after $t$ years is $C(t) = 100\,000 \times 1.04^{t}$.

1. a) Set up the exponential equation that determines how many years must pass for the capital to reach $\$160\,000$.

1. b) Solve it using logarithms and give the result rounded to the nearest month.

1. c) Without doing the full calculation, reason whether with an interest rate of $2 \%$ the time required would be exactly double. Check your conjecture with calculations.

A bacterial colony grows at a cumulative rate of $15 \%$ per day. If initially there are $2,000$ bacteria:

1. a) Write the expression for the number of bacteria after $t$ days and set up the inequality that determines when the population will exceed $10,000$ bacteria.

1. b) Solve it using logarithms and deduce the minimum number of full days that must pass.

Solve the logarithmic system:

Check that the solution satisfies both equations and that the values are valid for the logarithms.

Solve the exponential system:

Hint: convert each equation into a linear equation for the exponents.

Solve the logarithmic equation $\log_{2} x + \log_{4} x = 3$.

1. a) Express $\log_{4} x$ in terms of $\log_{2} x$ using the change of base formula.

1. b) Solve the resulting equation and check the solution.

Solve the equation $4^{x} - 3 \times 2^{x + 1} + 8 = 0$.

1. a) Rewrite the equation using only powers of base $2$ and substitute $t = 2^{x}$.

1. b) Solve the quadratic equation in $t$.

1. c) Reverse the substitution and check the two solutions.