{"id":11234,"date":"2015-07-14T19:07:55","date_gmt":"2015-07-14T19:07:55","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11234"},"modified":"2015-09-09T21:40:21","modified_gmt":"2015-09-09T21:40:21","slug":"solve-applied-problems-involving-exponential-and-logarithmic-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/solve-applied-problems-involving-exponential-and-logarithmic-equations\/","title":{"raw":"Solve applied problems involving exponential and logarithmic equations","rendered":"Solve applied problems involving exponential and logarithmic equations"},"content":{"raw":"
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In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.<\/p>\r\n

One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life<\/strong>. The table below\u00a0lists the half-life for several of the more common radioactive substances.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Substance<\/th>\r\nUse<\/th>\r\nHalf-life<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
gallium-67<\/td>\r\nnuclear medicine<\/td>\r\n80 hours<\/td>\r\n<\/tr>\r\n
cobalt-60<\/td>\r\nmanufacturing<\/td>\r\n5.3 years<\/td>\r\n<\/tr>\r\n
technetium-99m<\/td>\r\nnuclear medicine<\/td>\r\n6 hours<\/td>\r\n<\/tr>\r\n
americium-241<\/td>\r\nconstruction<\/td>\r\n432 years<\/td>\r\n<\/tr>\r\n
carbon-14<\/td>\r\narcheological dating<\/td>\r\n5,715 years<\/td>\r\n<\/tr>\r\n
uranium-235<\/td>\r\natomic power<\/td>\r\n703,800,000 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:<\/p>\r\n\r\n

[latex]\\begin{cases}A\\left(t\\right)={A}_{0}{e}^{\\frac{\\mathrm{ln}\\left(0.5\\right)}{T}t}\\hfill \\\\ A\\left(t\\right)={A}_{0}{e}^{\\mathrm{ln}\\left(0.5\\right)\\frac{t}{T}}\\hfill \\\\ A\\left(t\\right)={A}_{0}{\\left({e}^{\\mathrm{ln}\\left(0.5\\right)}\\right)}^{\\frac{t}{T}}\\hfill \\\\ A\\left(t\\right)={A}_{0}{\\left(\\frac{1}{2}\\right)}^{\\frac{t}{T}}\\hfill \\end{cases}[\/latex]<\/div>\r\n

where<\/p>\r\n\r\n