Exercises: Logarithmic functions

Calculate the value of the following logarithms by applying the definition directly, without using a calculator:

a) $\log_2 32$

b) $\log_3 \frac{1}{9}$

c) $\log 10000$

Find the value of the unknown $x$ in each of the following equations using the definition of a logarithm:

a) $\log_x 25 = 2$

b) $\log_5 x = 3$

c) $\log_2 64 = x$

Determine the domain of definition of the following logarithmic functions:

a) $f(x) = \log(x - 5)$

b) $g(x) = \ln(2x + 8)$

State, giving reasons for your answer and without sketching them, which of the following functions are increasing and which are decreasing:

a) $f(x) = \log_3 x$

b) $g(x) = \log_{0,5} x$

c) $h(x) = \ln x$

Use a calculator and the change of base formula to find the value of $\log_4 15$. Round the answer to three decimal places.

Look at the graphs of the logarithmic functions shown in the image above. Match, giving reasons for your answer, Function A and Function B with their respective algebraic expressions:

1) $y = \log_2 x$

2) $y = \log_{1/2} x$

Knowing that, approximately, $\log 2 = 0.301$, calculate the value of $\log 8$ using only the laws of logarithms.

Expand the following expression by applying the laws of logarithms (logarithm of a product, quotient and power), so that only logarithms of single variables remain:

$\log \left( \frac{x^3 \times y^2}{\sqrt{z}} \right)$

Combine the following expression into a single logarithm using the properties of logarithms:

$2 \log a + \log b - 3 \log c$

Solve the following simple logarithmic equations:

a) $\log(3x - 1) = \log(x + 7)$

b) $\log_2(x - 4) = 3$

Given the function $f(x) = \log_3(12 - 3x)$:

a) Determine its domain algebraically.

b) State the equation of its vertical asymptote.

c) In which direction (left or right) does the graph of this function open relative to its asymptote?

Capital is invested in a bank fund offering annual compound interest. The formula governing the accumulated capital $C$ over time $t$ in years is $C = C_0 \times 1.04^t$, where $C_0$ is the initial capital. If $40\,000$ pesos are invested, determine how many full years must pass for the capital to exceed $60\,000$ pesos. Use logarithms to solve for the unknown.

From the graph of the function $f(x)$ shown above, which corresponds to a translated logarithm of the form $f(x) = \log_a (x + b)$:

a) State the value of $b$ by observing the vertical asymptote.

b) Use the point $B(1, 2)$ to calculate the base $a$ of the logarithm.

c) Write the full algebraic expression for the function.

Sound intensity is measured in decibels (dB) using a logarithmic function. The formula is $d = 10 \times \log\left(\frac{I}{I_0}\right)$, where $d$ is the decibel level, $I$ is the sound intensity and $I_0$ is the threshold of hearing intensity (a constant). Rearrange for the intensity $I$ in terms of the decibels $d$ and $I_0$.

Knowing that $\log 2 = 0.301$ and $\log 3 = 0.477$, calculate the value of $\log 12$ and $\log 5$ using the laws of logarithms, without directly using the log button on the calculator for those numbers. (Hint: remember that $5 = \frac{10}{2}$).

Solve the following logarithmic equation, checking the validity of the solutions obtained:

$\log(x + 2) + \log(x - 1) = 1$

Find the values of $x$ and $y$ that satisfy the following system of non-linear equations:

$\begin{cases} x - y = 9 \\ \log x - \log y = 1 \end{cases}$

The Richter scale for measuring earthquake magnitude is logarithmic and is defined as $M = \log\left(\frac{A}{A_0}\right)$, where $A$ is the amplitude of the seismic wave and $A_0$ is a reference amplitude.

An earthquake in city Alpha had a magnitude of $6$ on the Richter scale, and another in city Beta had a magnitude of $8$. How many times greater was the amplitude of the seismic wave of the Beta earthquake compared to that of Alpha? Show your reasoning mathematically.

Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x - 5)$. Express the result in interval notation.

Carbon-14 dating is used to date ancient organic remains. The proportion of C-14 decreases exponentially over time according to the function $P(t) = 100 \times 0.5^{\frac{t}{5730}}$, where $P$ is the percentage of C-14 remaining relative to the original and $t$ is the time elapsed in years.

If a fossil is discovered that contains only $15\%$ of its original Carbon-14, what is the estimated age of the fossil? Express the result in years and round to the nearest whole number.

The image shows the graph of the exponential function $f(x) = e^x$ and its inverse function $g(x)$.

a) Write the algebraic expression for $g(x)$.

b) Explain the geometric meaning of the dashed orange line $y = x$ in relation to both curves.

c) Determine the domain and range of both functions, justifying how they swap between a function and its inverse.

It is known that a logarithmic function of the form $f(x) = \log_a(x + b)$ passes through the origin $(0, 0)$ and through the point $(6, 1)$.

Set up a system of equations to find the values of the parameters $a$ and $b$. Once found, write the final algebraic expression for the function and state its vertical asymptote.

In computer science, the execution time $T$ of a sorting algorithm depends on the amount of data $n$ to process, and follows a function of the type $T(n) = k \times n \times \log_2 n$, where $k$ is a constant specific to the computer.

If when processing $n = 8$ items of data the computer takes $T = 24$ milliseconds:

a) Calculate the value of the constant $k$.

b) How long will the same computer take to process $n = 32$ items of data?