Reasoning: In scientific notation, the coefficient must be $\geq 1$ and $< 10$.

Reasoning: Reference position is after the first non-zero digit $(5$ in $053.7)$.

Reasoning: Log base cannot be $1\ (\log_{\ 10}$ is undefined when $base = 1)$.

Reasoning: Same base logs. $\Longrightarrow$ Equate arguments:
$x + 3 = 15x - 4$ $\Longrightarrow$ $x = 0.5$

Reasoning: $\log_{7}\ 7^{- 3} = - \ 3$ ; $\log_{2}\ 4^{3} = \log_{2}\ 2^{6} = 6$
So, $Total = 3$

Reasoning: $0.00327 = 3.27 \times 10^{- 3}$, so characteristic $= - \ 3$.

Reasoning: Log of sum has no law: $log\ (M + N)$ is not separable.

Reasoning: Power rule: $\log_{b}$ $(g^{h}) = h\ \log_{b}\ g$

Reasoning: Quotient rule of logarithms.

Reasoning: $100^{2} = 10^{4}$, base $\sqrt{10}=10^{1/2}$. So, $4 \div 1/2 = 8$

Reasoning: $18 = 2 \times 3^{2}$, apply product & power rules.

Reasoning: Combine logs: $\log$ $\frac{5\ \times \ 3}{8\ \times \ 2}$

Reasoning: $(100^{0} = 1)$, so $\log_{10}\ 1 = 0$

Reasoning: $6.25$ is already between $1$ and $10$, so power of $10$ is $(10^{0} = 1)$.

Reasoning: Base $(e)$ (natural log) is irrational.

Reasoning: $(\sqrt{x})^{4}=25$ $\Longrightarrow$ $x^{2} = 25$, valid base $\Longrightarrow$ $(x = 5)$

Reasoning: $\log_{\ }\ 10^{4} \div \ \log_{\ }\ 10 = {log\ }_{10}\ 10^{4} = 4$

Reasoning: $5\ log\ 2 - 2\ log\ 5 = log\ (32) - log\ (25)$
rom identity, $x \times y \times z = 1$