Exercises: 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) $10^{x} = 0,001$

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 the equations:

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

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

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) $5^{2 - x} = 125$

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 by applying the appropriate inverse function:

1. a) $\ln x = 4$

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

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

1. d) $\log x = 2,5$

To solve the exponential equation $8^{x-1} = 4^{x+2}$, two students propose different paths:

• Paula states that the most direct way is to factor the bases into powers of $2$ ($8 = 2^3$ and $4 = 2^2$) and equate the resulting exponents.

• Sergio suggests applying common logarithms directly to both sides: $\log(8^{x-1}) = \log(4^{x+2})$, and bringing the exponents down using the power property of logarithms.

1. a) Solve the equation using the method proposed by Paula.
2. b) Explain why both methods are mathematically valid and argue which of the two is more advantageous and less prone to calculation errors in this particular case.

Lucía tries to solve the exponential equation $4^x - 3 \cdot 2^x - 4 = 0$ and hands in the following work in her notebook:

"Step 1: I make the substitution $t = 2^x$, so $4^x = (2^x)^2 = t^2$.

Step 2: I get $t^2 - 3t - 4 = 0$, whose solutions are $t_1 = 4$ and $t_2 = -1$.

Step 3: I undo the substitution:

• For $t_1 = 4 \implies 2^x = 4 \implies x = 2$.

• For $t_2 = -1 \implies 2^x = -1 \implies x = -0,5$ (because $2 \cdot (-0,5) = -1$).

Conclusion: The equation has two real solutions: $x = 2$ and $x = -0,5$".

1. a) Identify and explain in detail the conceptual error made by Lucía in Step 3, justifying through the properties of the exponential function why the equality $2^x = -1$ is impossible in the set of real numbers.
2. b) Correct the solution and determine the final set of valid solutions to the equation.

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 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 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.

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.

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.

Given the exponential equation $3^{2x-1} = 50$:

1. a) Without using a calculator or logarithms, bound the value of the unknown $x$. To do so, identify between which two consecutive integer powers of base $3$ the number $50$ lies, deduce between which two consecutive integers the exponent $2x-1$ must lie, and from there, obtain an interval bounded by two simple rational or decimal numbers for $x$.
2. b) Solve the equation exactly by applying common logarithms to both sides and solving for the unknown $x$.
3. c) Obtain the approximate numerical value to three decimal places and check whether your initial estimate from part a) was consistent.

According to the used vehicle valuation tables used by the Tax Agency in Spain, a combustion passenger car depreciates at an approximate rate of $12\,\%$ per year during its first years of circulation. A courier company purchases a delivery van for $25\,000\text{ \euro}$. Its residual value $V(t)$, expressed in euros after $t$ years of service, is modeled by the function $V(t) = 25\,000 \cdot (0,88)^t$.

1. a) Set up the exponential equation that allows calculating how many years must elapse for the value of the van to drop to exactly $10\,000\text{ \euro}$.
2. b) Solve this equation using logarithms, round the result to the nearest hundredth, and interpret the obtained value by stating the approximate years and months it takes to reach that value.
3. c) If the company had purchased a larger capacity van for $50\,000\text{ \euro}$ with the same depreciation rate of $12\,\%$ per year, would it take more time, less time, or the same amount of time to lose exactly half of its initial value? Justify your answer from the algebraic structure of the equation, without needing to repeat all the calculations.

Solve the exponential equation $9^{x} - 4 \cdot 3^{x} + 3 = 0$ 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 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.

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

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.

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 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).

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.

Design an exponential equation with a quadratic structure of the form $a \cdot 4^x + b \cdot 2^x + c = 0$ that simultaneously satisfies the following three conditions:

1. The coefficients $a$, $b$, and $c$ are integers and $a = 1$.

2. The real number $x = 3$ is the only real solution of the equation.

3. When applying the substitution $t = 2^x$, the quadratic equation $t^2 + bt + c = 0$ has two distinct real solutions, but one of them does not produce any real solution for $x$.

1. a) Deduce the integer values of $b$ and $c$ that satisfy these conditions, explaining the reasoning that connects the roots of the quadratic equation in $t$ with the range of the exponential function $2^x$.
2. b) Solve the resulting equation step by step to prove that it indeed only admits $x = 3$ as a real solution in $\mathbb{R}$.

Consider the following system of non-linear equations:

$\begin{cases} \log x + \log y = 2 \\ x - y = 15 \end{cases}$

1. a) Solve the system algebraically by applying the properties of logarithms and the substitution method.
2. b) When solving the quadratic equation that arises in the process, two pairs of values are obtained: $(20, 5)$ and $(-5, -20)$. Explain analytically, relying on the definition and the domain of validity of the original logarithmic function, why one of them must be discarded.
3. c) If a student replaces the first equation with $\log(x \cdot y) = 2$, transforming the system into:

$\begin{cases} \log(x \cdot y) = 2 \\ x - y = 15 \end{cases}$

is this new system equivalent to the original one? Compare with justification the domains of the expressions $\log x + \log y$ versus $\log(x \cdot y)$ and determine whether the pair $(-5, -20)$ is an admissible solution to this second system.

Topic challenge. Your challenge is to prepare a viable financial report for a youth micro-cooperative using data from the INE (median salary and average household expenditure).

Stage 1 (Data and Gauss): Look up the average monthly expenditure across three categories (housing, transport, and food) on the INE. Set up a $3 \times 3$ system with budget constraints and a fixed savings target, and solve it using Gaussian elimination to determine the allocations.

Stage 2 (Polynomials and viability): Model costs and revenues using polynomials. Factor them and construct the average cost function as an algebraic fraction. Set up and solve a quadratic inequality to establish the break-even point and a system of inequalities to define the feasible region.

Stage 3 (Long-term projection): Model the reserve fund against inflation (average rate of $2,5\%$) in GeoGebra using a non-linear system with exponential and logarithmic functions. Study its end behaviour ($t \to +\infty$) to predict whether the project will collapse or stabilise.

Submit a brief executive report with GeoGebra screenshots and your justified proposal.

Assessment criteria:

• Correctness in the application of Gaussian elimination, factorisation, and solving inequalities and systems.

• Rigour in the analysis of end behaviour and its socioeconomic interpretation.

• Coherent integration of real data from the INE into the final report.