Logarithms

Simplify $\log_{b} \sqrt[3]{b}$.

Evaluate $\log_{2} 64 - \log_{2} 32$.

What is the value of $\log_{10} (1/100)$?

Find $x$ if $\log_{4} x = 0.5$.

If $a^2 + b^2 = 7ab$, find the value of $\log \left(\frac{a+b}{3}\right)$.

Find the value of $\log_{2} \left(\frac{1}{32}\right)$.

Calculate $\log_{5} (0.04)$.

Solve for $x$ if $\log_2(x-1) + \log_2(x-3) = 3$.

Find $x$ if $\log_x 2 + \log_x 4 + \log_x 8 = 12$.

Evaluate the nested expression: $\log_2 (\log_2 (\log_3 (\log_3 27^3)))$.

What is the value of $\log_{2} 128$?

If $\log_{x} 25 = 2$, then $x$ is:

Find $x$ if $\log_{9} 3 = x$.

What is the value of $\log_{10} 10^5$?

Evaluate the value of $\log_{3\sqrt{3}} (243)$.

Evaluate $\log_{k} k^n$.

Find $x$ if $\log_{10} (x^2) = 4$. Assume $x > 0$.

What is the value of $\log_{\sin 30^\circ} 0.25$?

Given $\log_2 a = p$ and $\log_8 b = q$, express $\log_4 (ab)$ in terms of $p$ and $q$.

Solve: $2^{\log_2 (x-3)} = 5$.

Find the value of $\log_{10}(0.1^{-1})$.

Evaluate $\log_{10}(10^3 \times 10^2)$.

What is $\log_{10}(10 \div 1000)$?

If $10^x = 10000000$, find $x$.

Evaluate the value of $\log_{10} \left( \frac{\sqrt[3]{10}}{0.001} \right)$.

Simplify the fraction $\frac{\log_{10} 1000}{\log_{10} 100}$.

If $\log_{10} 2 \approx 0.301$ and $\log_{10} 3 \approx 0.477$, find $\log_{10} 6$.

If $\log_{10} 5 = a$, then what is the value of $\log_{10} 20$ in terms of $a$?

Find the value of $x$: $\log_{10} (x^2 - 1) = \log_{10} 3 + \log_{10} 8$.

Solve for $x$: $10^{x+2} - 10^x = 990$.

Evaluate the product $\log_3 5 \times \log_5 9$.

Find the value of $x$ for which $\log_7 x = 0$.

Evaluate $\log_{10} 100 + \log_{10} 0.1$.

Simplify $5 \log_a x - 3 \log_a x$.

If $\log_7 (2x - 1) = \log_7 5$, what is the value of $x$?

Solve for $x$: $\log_{10} 2 + \log_{10} (x + 3) = 1$.

Evaluate $\log_2 8 + \log_3 9 + \log_5 25$.

Solve the equation: $\log_{10}(x+1) + \log_{10}(x-1) = \log_{10} 3$.

If $x = \log_2 3$ and $y = \log_3 5$, express $\log_2 15$ in terms of $x$ and $y$.

Evaluate $\log_2 \log_3 \log_2 512$.

Evaluate $\log_{10} (10^4)^{1/2}$.

What is the value of $\log_{10} (2 \times 5 \times 10)$?

Find $n$ if $\log_{10} 100^n = 6$.

The value of $\log_{10} 0.2 + \log_{10} 50$ is:

If $x^2 + y^2 = 14xy$, which of the following is equal to $2 \log_{10} \left(\frac{x+y}{4}\right)$?

Simplify $\log_{10} \left(\frac{a^2}{bc}\right) + \log_{10} \left(\frac{b^2}{ca}\right) + \log_{10} \left(\frac{c^2}{ab}\right)$.

What is the characteristic of $\log_{10} 0.000034$?

If $\log_{10} a = 1.2$ and $\log_{10} b = 0.6$, find the value of $\log_{10} (a \sqrt{b})$.

Solve for $x$: $\log_{10} (3x^2 + 7) = 2$

Find the value of $x$ satisfying: $\log_{10} (x+1) + \log_{10} (x-1) = \log_{10} 11 + 2\log_{10} 3$

What is the value of $\log_{6} 36 - \log_{6} 6$?

If a solution's $[H^+]$ increases by a factor of 10, the pH will:

The relationship between a base $b$, an exponent $y$, and a number $x$ is $b^y = x$. This is written as:

What is $\log_{10} 100^{3}$?

Evaluate $\log_7 1$.

In cooling, Newton's Law can be expressed as $t = -k \log_{10}(\frac{T - T_s}{T_0 - T_s})$. If $\frac{T - T_s}{T_0 - T_s} = 0.1$, what is the value of $t$?

If $\log_{10} 5 = z$, then find $\log_{10} 125$ in terms of $z$.

If $a, b, c$ are distinct positive numbers such that $(\log b \cdot \log c) + (\log a \cdot \log b) + (\log a \cdot \log c) = 0$, find the value of $a^{\log(bc)} \cdot b^{\log(ac)} \cdot c^{\log(ab)}$.

Evaluate the sum: $\sum_{n=1}^{89} \log_{10} (\tan n^\circ)$.

If $\log_k x \cdot \log_5 k = 3$, find $x$.

Solve the equation: $\log_{5} (x^{2}) = 4$ (given $x > 0$)

Solve for $x$: $\log_{2} (\frac{x}{2}) = 4$

Find $x$ in $\log_{x} 243 = 5$.

If $\log_{4} x = 3$, then $x$ is:

Find $x$ in the equation $\log_{x-1} 4 = 2$.

Solve for $x$: $\log_2(\log_3(\log_4 x)) = 0$.

Determine $x$ such that $\log_x(\frac{1}{3}) = -\frac{1}{2}$.

Solve for $x$: $\log_2 x + \log_4 x = \log_2(3\sqrt{3})$.

Solve for $x$: $\log_{10} x + \log_{10} x^2 + \dots + \log_{10} x^{10} = 110$.

Determine $x$ if $x^{\log_5 x} = 625$.