Logarithms

Calculate the value of the following logarithms by directly applying the definition:

• $\log_2 64$

• $\log_3 \frac{1}{9}$

• $\log_{10} 0.001$

• $\log_5 \sqrt{5}$

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

• $\log_x 81 = 4$

• $\log_2 x = 5$

• $\log_7 x = -2$

Knowing that $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$, calculate the approximate numerical value of $\log 6$ and $\log 1.5$ using the properties of logarithms.

Simplify the following mathematical expression to a single logarithm and calculate its numerical value:

$\log 5 + \log 20 - \log 10$

Apply the properties of logarithms in reverse to express the following combination as a single logarithm:

$3 \log x + \frac{1}{2} \log y$

Expand the following logarithmic expression by applying all possible properties, so that it is expressed in terms of $\log x$, $\log y$ and $\log z$:

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

Combine the following algebraic expression into a single logarithm, applying the corresponding properties:

$2 \log a - 3 \log b + \log c - \frac{1}{3} \log d$

Knowing only that $\log 2 \approx 0.301$, calculate the approximate value of $\log 25$ and $\log \sqrt[3]{0.02}$ without entering the direct operation into the calculator, relying only on the properties of logarithms.

Use the change-of-base formula to calculate the value of $\log_5 120$ using only the common logarithm ($\log$) or natural logarithm ($\ln$) key on your calculator. Round the result to three decimal places.

Simplify the following combined logarithmic expression and calculate its exact numerical value without using a calculator:

$\frac{\log_2 8 + \log_3 81 - \log 1000}{\log_5 125 \times \log_4 \sqrt{4}}$

Prove, using the change-of-base formula and the fundamental properties of logarithms, that $\log_a b \times \log_b a = 1$ for any numbers $a, b > 0$ and different from $1$. Once proven, apply this result to mentally calculate the value of $\log_8 2$, knowing beforehand that $\log_2 8 = 3$.

The amount of Carbon-14 ($C-14$) in an organic archaeological remain follows the law of radioactive decay $N(t) = N_0 \times e^{-\lambda t}$, where the decay constant is $\lambda \approx 0.0001209$ and $t$ is the elapsed time in years. If laboratory analysis detects that only $30\,\%$ of its original amount of $C-14$ remains in a fragment of prehistoric bone (that is, $N(t) = 0.30 \times N_0$), what is the estimated age of the bone?

Calculate the value of $x$ in the exponential equation $3^x = 45$. Express the final result rounded to two decimal places (you may use a calculator for the final step).

Solve the logarithmic equation $\log_3 (2x + 5) = 2$ by applying the definition of a logarithm to rearrange for the unknown $x$.

A principal of $100\,000$ pesos is deposited in a bank at a compound interest rate of $3.5\,\%$ per year. How many full years must elapse for the accumulated amount to exceed $140\,000$ pesos?

Remember that the formula for the final amount is $C_f = C_i \times \left(1 + \frac{r}{100}\right)^t$.

A population of bacteria doubles every hour. If there are initially $1500$ bacteria, the number of bacteria after $t$ hours is given by the function $P(t) = 1500 \times 2^t$. How much exact time must pass for the population to reach $100000$ bacteria?

A new vehicle has an initial cost of $480\,000$ pesos. Due to market wear and tear, its value depreciates by $12\,\%$ each year. The function describing its residual value after $t$ years is $V(t) = 480\,000 \times (0,88)^t$. How many years must pass for the value of the car to be reduced to exactly half of its original price?

Solve the exponential equation $5^{x+1} = 2^{3x-2}$ by taking common logarithms of both sides and rearranging for the unknown $x$. Express the final result rounded to three decimal places.

On a well-known social network, the number of followers of an account with viral content grows according to an exponential model $S(t) = S_0 \times k^t$, where $t$ is measured in days. It is known that on the initial day ($t = 0$) the account had $200$ followers and on day $5$ it reached $6400$ followers. First, calculate the exact value of the growth constant $k$. Then, determine on which exact day it will pass the one-million-follower mark.

Solve the following system of logarithmic equations, finding the values of $x$ and $y$:

The following graph represents the population growth of two bacterial colonies, A and B, in a laboratory. The scale of the vertical axis (which indicates the number of bacteria $N(t)$) is logarithmic in base 10, while the horizontal axis (time in hours) has a linear scale. By observing the line describing the behavior of colony A, find its associated exponential function $N(t) = N_0 \times 10^{k \times t}$, knowing that it passes exactly through the points $(0, 10^2)$ and $(4, 10^4)$. At what time $t$ will colony A reach exactly one million bacteria ($10^6$)?

The pH of a chemical solution is defined by the formula $pH = -\log [H^+]$, where $[H^+]$ represents the concentration of hydrogen ions measured in mol/L. If a sample of lemon juice has a hydrogen ion concentration of $10^{-3}$ mol/L, what is its pH value?

The magnitude of an earthquake on the Richter scale is calculated using the formula $M = \log \left( \frac{A}{A_0} \right)$, where $A$ is the maximum amplitude of the seismic waves and $A_0$ is a standard reference amplitude. If a particular earthquake records an amplitude $1000$ times greater than $A_0$, what is its magnitude on the Richter scale? Furthermore, determine how many times greater the amplitude of an earthquake of magnitude $6$ is compared to one of magnitude $4$.

The sound intensity level, measured in decibels (dB), is calculated using the physics formula $L = 10 \times \log \left( \frac{I}{I_0} \right)$, where $I$ is the sound intensity in $\text{W/m}^2$ and $I_0 = 10^{-12} \text{ W/m}^2$ represents the minimum threshold of human hearing. If the heavy traffic noise on an avenue records a level of $85$ dB, what is its sound intensity $I$? If a jackhammer on a construction site emits twice the sound intensity of that traffic, how many exact decibels will the sound level meter record?