Brain Teaser 2: Find the value of e^x, where x = 4 + ln 2

If you can solve the above, you should be able to solve these too (without using calculator):

Find the value of:
(a) 3^x, where x = 4 + log₃ 2. (My question)
(b) 2^z, where z = 5 + log₂ 3  (IGCSE Cambridge Add-Maths 0606, Yr 2010/Summer/p22, Q10(b))

I will post their answers in a week or so. Meanwhile, you may post your answers by way of comments to this post. Have fun with maths!

------------------------------------------------------------------------------------------------
29.04.2014

My Proposed Solutions to These Questions:

Find the value of:

 (Lead Question): e^x, where x = 4 + ln 2
(a) 3^x, where x = 4 + log₃ 2. (My question)
(b) 2^z, where z = 5 + log₂ 3  (IGCSE Cambridge Add-Maths 0606, Yr 2010/Summer/p22, Q10(b))

---------------------------------------------------------------------------------

For the lead question:

Since            x = 4 + ln 2,

therefore:     e^x = e^{(4 + ln 2)} = e⁴ x e^{ln 2} ..............Law of index: b^{(m + n)} = b^m x bⁿ

Now:           let y = e^{ln 2}

ln both sides: ln y = ln 2

therefore:          y = 2 = e^{ln 2} (Hence, remember: e^{ln a} = a; 10^{lg b} = b; etc.)

Hence, e^x = e^{(4 + ln 2)} = e⁴ x e^{ln 2} = 2e⁴ (Answer)

---------------------------------------------------------------------------------

(a)    Since           x = 4 + log₃ 2

therefore: 3^x = 3^(4 + log₃ 2) = 3⁴ x 3^log₃ 2 = 3⁴ x 2 (see above)

                  3^x = 81 x 2 = 162 (Answer)

(b)   Since         z = 5 + log₂ 3

therefore:  2^z = 2^(5 + log₂ 3) = 2⁵ x 2^log₂3 = 2⁵ x 3

                2^z   = 32 x 3 = 96 (Answer)

---------------------------------------------------------------------------------

So, remember:

·        e^{ln a} = a

·        10^{lg b} = b

·        b^log_b c = c

Have fun with maths!

--------------------------------------------------------------------------------
(1.5.2014)

Brain Teaser 2B (requires slightly more skill than Brain Teaser 2)

Find the exact value of:

1) 3^x , where x = 2 + log₉ 3

2) 2^y , where y = 3 + log₈ 2

-------------------------------------------------------------------------------