Question

Determine whether each of the following statements is true or false:

(i) $\ln \left(a+b\right)=\ln \left(ab\right)$

(ii) $\ln \left(a^x\right)=x\ln \left(a\right)$

(iii) $\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

(iv) $e^{\frac{a}{b}}=e^a-e^b$

(v) $e^{\ln \left(x\right)}=x$

(vi)$\ln \left(e^x\right)=1$

Short answer

(i) True

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

Step-by-step solution

Step 1. Given Information

(i) $\ln \left(a+b\right)=\ln \left(a\right)+\ln \left(b\right)$

(ii) $\ln \left(a^x\right)=x\ln \left(a\right)$

(iii) $\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

(iv) $e^{\frac{a}{b}}=e^a-e^b$

(v) $e^{\ln \left(x\right)}=x$

(vi)$\ln \left(e^x\right)=1$

Step 2. Determine True or False.

(i) $\ln \left(a+b\right)=\ln \left(a\right)+\ln \left(b\right)$

True as it is a logarithmic formula.

(ii) $\ln \left(a^x\right)=x\ln \left(a\right)$

True as it is a logarithmic formula.

(iii) $\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)$

True as it is a logarithmic formula.

(iv)$e^{\frac{a}{b}}=e^a-e^b$

False as $e^{\frac{a}{b}}=\frac{e^a}{e^b}$.

(v) $e^{\ln \left(x\right)}=x$

True.

$$\begin{gathered} y=e^{\ln \left(x\right)} \\ \ln \left(y\right)=\ln \left(e^{\ln \left(x\right)}\right) \\ \ln \left(y\right)=\ln \left(x\right)\ln \left(e\right) \\ \ln \left(y\right)=\ln \left(x\right).1 \\ y=x \end{gathered}$$

(vi)$\ln \left(e^x\right)=1$

False.

$$\begin{gathered} \ln \left(e^x\right)=1 \\ \ln \left(e^x\right)=x\ln \left(e\right) \\ =x \end{gathered}$$