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 $4^{x} - 5 \cdot 2^{x} + 4 = 0$ 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 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?

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.

A family deposits $5.000\ \text{ €}$ into a savings account at $4 \%$ annual compound interest, so that the capital after $t$ years is $C(t) = 5.000 \cdot 1,04^{t}$.

1. a) Set up the exponential equation that determines how many years must pass for the capital to reach $8.000\ \text{ €}$.

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.

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.