Solve 12=14.7e^-0.0004h

$$12 = 14.7 e ^ { - 0.0004 h }$$

Solve for h

$$h=\ln(\frac{7^{5000}}{1412467032139426036835209667016147333668896175184541116813688085857118169842707512558089126316711526373356032084313660827642038380699793383359711857266399234310517778518653990118779996451317070693734982126313237525531112153728440359509005359548607334184534055755667368015655874054646996404990508496994723579009056175713766182282164342131815209915566771264986517822041740618309392391768613413832940182402258386927255961470051442432810752756294953390938131989667356336063296910238424541258358886568731339812872409800088380736682218042644329108940307890202194405781984882673397682388722799021574203072475705104238458688725967358918058187277964357530185180866413560128513025467268230092502183280182519073402454498631832656379878621985110463629854619495872811191399072280043859428809539588165545676252960869168857748289344499413624165886753269403325611036645569826222068344742198110818724049295034819913767403798259987914118798027175838854985751152994717434692411170702303981033786152327937102909926564448428955118303557331520208041579200900418119518804567055154683494461827317423276859892776076207095258783187664883683489650154749978641197654414333569280123441117657353363935578792149370043475682086659587177640592935928875142928435570470891648764831166156918862038129975556901718921697337552244690324750787978309013215799401273372106943772834399222802740607982347867404348934581201983411010338125067200466098911607002840021009804529640397887043353026193375978620521922803714811321641471865141690909171919093760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000})$$

Steps Using Definition of Logarithm

Divide both sides by 14.7.

$$\frac{12}{14.7}=e^{-0.0004h}$$

Expand \frac{12}{14.7} by multiplying both numerator and the denominator by 10.

$$\frac{120}{147}=e^{-0.0004h}$$

Reduce the fraction \frac{120}{147} to lowest terms by extracting and canceling out 3.

$$\frac{40}{49}=e^{-0.0004h}$$

Swap sides so that all variable terms are on the left hand side.

$$e^{-0.0004h}=\frac{40}{49}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.0004h})=\log(\frac{40}{49})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$-0.0004h\log(e)=\log(\frac{40}{49})$$

Divide both sides by \log(e).

$$-0.0004h=\frac{\log(\frac{40}{49})}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$-0.0004h=\log_{e}\left(\frac{40}{49}\right)$$

Multiply both sides by -2500.

$$h=\frac{\ln(\frac{40}{49})}{-0.0004}$$

Solve for h (complex solution)

$$h=-i\times 5000\pi n_{1}+\ln(\frac{7^{5000}}{1412467032139426036835209667016147333668896175184541116813688085857118169842707512558089126316711526373356032084313660827642038380699793383359711857266399234310517778518653990118779996451317070693734982126313237525531112153728440359509005359548607334184534055755667368015655874054646996404990508496994723579009056175713766182282164342131815209915566771264986517822041740618309392391768613413832940182402258386927255961470051442432810752756294953390938131989667356336063296910238424541258358886568731339812872409800088380736682218042644329108940307890202194405781984882673397682388722799021574203072475705104238458688725967358918058187277964357530185180866413560128513025467268230092502183280182519073402454498631832656379878621985110463629854619495872811191399072280043859428809539588165545676252960869168857748289344499413624165886753269403325611036645569826222068344742198110818724049295034819913767403798259987914118798027175838854985751152994717434692411170702303981033786152327937102909926564448428955118303557331520208041579200900418119518804567055154683494461827317423276859892776076207095258783187664883683489650154749978641197654414333569280123441117657353363935578792149370043475682086659587177640592935928875142928435570470891648764831166156918862038129975556901718921697337552244690324750787978309013215799401273372106943772834399222802740607982347867404348934581201983411010338125067200466098911607002840021009804529640397887043353026193375978620521922803714811321641471865141690909171919093760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000})$$

$$n_{1}\in \mathrm{Z}$$