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Evaluate log_(27)sqrt(54)-log_(27)sqrt(6). |
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Answer» Solution :If you can enter base 27 into your calculator, enter the entire expression to get the answer `0.bar(3)=(1)/(3)`. Otherwise, you can use the change-of-base formula to substitute `(log54)/(log27)" for "log_(27)sqrt(54)and(logsqrt(6))/(log27)" for"log_(27)sqrt(6)`. If you your logarithms well, you use properties to see that `log_(27)sqrt(54)-log_(27)sqrt(6)=log_(27)sqrt((54)/(6))` `=log_(27^(3))` `=(1)/(3)` The GRAPHS of all exponential functions `y=b^(x)` have roughly the same shape and pass through point (0,1). If `bgt1`, the graph increases as x increases and APPROACHES the x-axis as an asymp- tote as x decreases. The amount of curvature BECOMES greater as the value of b is made greater. If `0ltblt1`, the graph increases as x decreases and approaches the x-axis as an asmptote as x increases. The amount of curvature becomes greater as the value of b is made closer to zero.  The graphs of all logarithmic functions `y=log_(b)x` have roughly the same shape and pass through point (1,0). If `bgt1`, the graph increases as x increases and approaches the y-axis as an asymptote as x approaches zero. The amount of curvature becomes greater as the value of b is made greater. If `0ltblt1`, the graph decreases as x increases as x increases and approaches the y-axis as an asymptote as x approaches zero. The amount of curvature becomes greater as the value of b is made closer to zero.  Two numbers serve as special bases of exponentil functions The number 10 is convenient as a base because integer power of ten determine place values : for instance, `10^(2)=100 and 10^(-3)=0.001`. The inverse of `10^(x)` is `log_(10)x`. By convention, the base is not written when it is 10, so the inverse of `10^(x)` is simply written as log 10. The other special base is the number `e~~2.718281828 . . . . ` This is a nontermination and nonrepeating decimal. The inverse of `e^(x)` is `log_(e)x`, which is further abbreviated ln x, the NATURAL logarithm of x. On the TI - 84 graphing calculator, the command log will evalute the logarithm with respect to any base by entering the value whose log is being sought followed by the base. Log base e (ln) can also entered the value whose log is being sought followed by the base. Log base e (ln) can also be entered directly. Exponential growth is an important application of exponential functions Exponential growth reflects a CONSTANT rate of change from one time period to the next. |
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