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Question Number 207250 by hardmath last updated on 10/May/24

$$\mathrm{If}{2}^{\mathbf{a}}=5,3^{\mathbf{b}}=9\mathrm{and}{25}^{\mathbf{c}}=8$$$$\mathrm{Find}:\mathrm{a}\cdot \mathrm{b}\cdot \mathrm{c}=?$$

Answered by A5T last updated on 10/May/24

$$a={log}_{2}5;b=2;c={log}_{25}8=\frac{1}{2}{log}_{5}8={log}_5\left(2\sqrt{2}\right)$$$$\Rightarrow abc=2{log}_{2}5\times {log}_{5}2\sqrt{2}={log}_{2}8=3$$

Commented by hardmath last updated on 10/May/24

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}$$

Answered by Rasheed.Sindhi last updated on 11/May/24

$$3^{\mathbf{b}}=9\Rightarrow 3^{\mathrm{b}}=3^2\Rightarrow \mathrm{b}=2$$$$2^{\mathrm{a}}=5\Rightarrow 2=5^{\frac{1}{\mathrm{a}}}$$$${25}^{\mathrm{c}}=8\Rightarrow 5^{2\mathrm{c}}=2^3={\left(5^{\frac{1}{\mathrm{a}}}\right)}^3$$$$\Rightarrow 5^{2\mathrm{c}}=5^{\frac{3}{\mathrm{a}}}\Rightarrow 2\mathrm{c}=\frac{3}{\mathrm{a}}\Rightarrow \mathrm{ac}=\frac{3}{2}$$$$\left(\mathrm{ac}\right)\left(\mathrm{b}\right)=\frac{3}{2}\left(2\right)=3\Rightarrow \mathrm{abc}=3$$

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